Constructive Approaches to the Rigidity of Frameworks Viet Hang Nguyen

Constructive approaches to the rigidity of frameworks Viet Hang Nguyen

To cite this version: Viet Hang Nguyen. Constructive approaches to the rigidity of frameworks. Mathématiques générales[math.GM]. Universitéde Grenoble, 2013. Fran¸cais. .

HAL Id: tel-00992387 https://tel.archives-ouvertes.fr/tel-00992387 Submitted on 17 May 2014

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinéeau dépôtet àla diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiésou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche fran¸caisou étrangers,des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE Pour obtenir le grade de DOCTEUR DE L’UNIVERSITÉ DE GRENOBLE Spécialité : Mathématique et Informatique

Arrêté ministériel : 7 aoutˆ 2006

Présentée par Việt Hằng Nguyễn

Thèse dirigée par Zoltán Szigeti préparée au sein du Laboratoire G-SCOP et de l’École Doctorale MSTII

Constructive approaches to the rigidity of frameworks

Thèse soutenue publiquement le 17 octobre 2013, devant le jury composé de :

M. Jorge Ramírez Alfonsín Professeur, Université Montpellier 2, Président M. András Recski Professeur, Budapest University of Technology and Economics, Rapporteur M. Walter Whiteley Professeur, York University (Toronto), Rapporteur M. Victor Chepoi Professeur, Université de la Mediterranée (Marseille), Examinateur M. Tibor Jordán Professeur, Eotv¨ os¨ Loránd University (Budapest), Examinateur M. Zoltán Szigeti Professeur, Grenoble INP, Laboratoire G-SCOP, Directeur de thèse Acknowledgements

I would like to thank the members of the committee for their comments, suggestions on the manuscript as well as inspiring questions. I would like to express my deepest thank to my advisor Zoltán Szigeti for his constant support, encouragement, counsel and especially the great freedom I enjoyed during these three years. My special thank goes to András Seböfor his support and also for his energy which propagates in our research group and motivates us all. I am deeply grateful to Olivier Durand de Gevigney, my academic brother, co-author and true friend. I greatly appreciate his help and all the time we have shared. I would like also to thank my other collaborators Abdo Alfakih, Bill Jackson and Tibor Jordán for the joint works and many things I learned from them. I am thankful to all the people in our research group for discussions and sem- inars. I thank friends and staffs in the G-SCOP laboratory for having created a pleasant atmosphere. I would like to express my thank to my family in Vietnam for their love and encouragement. I am deeply grateful to my family-in-law and especially to my mother-in-law for her kindness and help. There are no words to express my thank to my husband Guillaume who has been and will be always with me in our journey of life. I am also thankful to our little angel Daniel Hai Phong, our unending source of joy and inspiration.

1 2 Contents

1 Introduction 7 1.1 Frameworks and rigidity ...... 10 1.2 Our contributions and organization of the thesis ...... 15

2 Preliminaries 19 2.1 Basic notions ...... 20 2.2 Graphs and digraphs ...... 20 2.3 Matroid theory ...... 23 2.4 Submodular functions ...... 26 2.5 Algebra and linear algebra ...... 27 2.6 Congruent motions and motions of rigid bodies ...... 30

3 Rigidity theory 33 3.1 Introduction ...... 34 3.2 Various types of rigidity ...... 35 3.2.1 Local rigidity ...... 35 3.2.2 Infinitesimal rigidity and rigidity matrix ...... 36 3.2.3 Local rigidity versus infinitesimal rigidity ...... 38 3.2.4 Static rigidity ...... 40 3.2.5 Infinitesimal rigidity versus static rigidity ...... 41 3.2.6 Global rigidity ...... 41 3.2.7 Universal rigidity ...... 42 3.2.8 Stress matrices ...... 42 3.3 Combinatorial rigidity ...... 44

3 Contents

4 Matroid approach 51 4.1 Introduction ...... 52 4.2 Characterizing abstract rigidity matroids ...... 53 4.3 1-extendable abstract rigidity matroids ...... 54 4.4 Submodular functions inducing ARMs ...... 58 4.5 A potential application ...... 61

5 Inductive constructions and decompositions 63 5.1 Introduction ...... 65 5.2 Graded sparse graphs ...... 68 5.2.1 Introduction ...... 68 5.2.2 A reduction theorem for graded sparse graphs ...... 70 5.2.3 Inductive construction of graded sparse graphs ...... 75 5.2.4 Decomposition of graded sparse graphs ...... 75 5.2.5 Graded sparse matroids ...... 81 5.3 (b, l)-sparse graphs ...... 86 5.3.1 Inductive construction ...... 86 5.3.2 (b, l)-pebble games ...... 93 5.4 Packing of matroid-based arborescences ...... 96 5.4.1 Introduction ...... 96 5.4.2 Proof of the main theorem ...... 102 5.4.3 Polyhedral description ...... 105 5.4.4 Algorithmic aspects ...... 106 5.4.5 Further remarks ...... 107

6 Inﬁnitesimal rigidity 109 6.1 Introduction ...... 110 6.2 Body-bar frameworks with boundary ...... 112 6.2.1 Body-bar frameworks ...... 112 6.2.2 Body-bar frameworks with bar-boundary ...... 115 6.3 Body-length-direction frameworks ...... 119 6.3.1 Introduction ...... 119 6.3.2 Frameworks with length-direction-rigid bodies ...... 120 6.3.3 Frameworks with direction-rigid bodies ...... 122 6.3.4 Frameworks with length-rigid bodies ...... 126 6.4 Direction-length frameworks ...... 131 6.4.1 Introduction ...... 131

4 Contents

6.4.2 0-extensions for direction-length graphs ...... 133 6.4.3 1-extensions for direction-length graphs ...... 135 6.4.4 Union of two spanning trees ...... 141

7 Global rigidity of direction-length frameworks 143 7.1 Introduction ...... 144 7.2 Quasi-generic direction-length frameworks ...... 145 7.3 Proof of Theorem 7.1.2 ...... 147 7.4 Further remarks ...... 156

8 Universal rigidity 159 8.1 Introduction ...... 160 8.2 Universal rigidity in R1 ...... 161 8.2.1 Universal rigidity of bipartite frameworks ...... 161 8.2.2 Bipartite graphs in general dimension ...... 164 8.2.3 Observations, open questions, conjectures ...... 165 8.3 Universal rigidity of tensegrity frameworks ...... 172 8.3.1 Introduction ...... 172 8.3.2 Preliminaries ...... 175 8.3.3 The conﬁguration matrix and Gale matrices ...... 177 8.3.4 Dominated tensegrity frameworks ...... 178 8.3.5 Aﬃnely dominated tensegrity frameworks ...... 180 8.3.6 Proof of main results ...... 183 8.3.7 Counter-example and conjectures ...... 185

Conclusion 189 References ...... 193

5 Contents

6 Chapter 1

Introduction

Contents 1.1 Frameworks and rigidity ...... 10 1.2 Our contributions and organization of the thesis . . . 15

7 La théorie de la rigiditése préoccupe originellement de la rigidité/flexibilitédes charpentes. Une charpente est un modèle mathématique décrivant une structure réelle. Une des premières sources de structures qui ont suscitél’étude de la rigidité vient de l’ingénierie des structures. Considérons une structure planaire constituée de barres solides jointes aux extrémités par des jonctions permettant àdeux barres jointes de bouger librement dans le plan autour du joint. Une question naturelle est de savoir si la structure peut être déformée. Par exemple, la structure planaire rectangulaire de la Figure 1.1 peut être déformée continûment en une structure planaire non-rectangulaire comme représentépar la figure en pointillés. Nous disons que la structure est flexible. Au contraire, la structure planaire de la Figure 1.2 ne peut pas être déformée continûment en une autre forme sans être cassée. Nous disons que la structure est rigide, ou plus précisément, localement rigide. Ce type de structures barres-et-joints planaires peut être modélisée par une paire (G, p) formée d’un graphe G = (V,E) et d’une application p : V → R2. Les sommets V de G correspondent aux joints et les arêtes E de G correspondent aux barres. L’application p détermine la position des joints. De la même manière, une structure barres-et-joints en dimension 3 (et plus généralement en dimension d) peut être modélisée par une application p : V → R3 (et p : V → Rd, respective- ment). Nous appelons cette paire une charpente barres-et-joints. Un déplacement continu d’une charpente (G, p) dans Rd est un déplacement continu des sommets de G dans Rd qui conserve les longueurs des arêtes. Une charpente est dite localement rigide si tous les déplacements continus de (G, p) préservent aussi la distance entre chaque paire de sommets de G. De manière équivalente, tout déplacement continu de (G, p) est la restriction d’un déplacement congruent de l’espace tout entier. Déterminer si une charpente est localement rigide est difficile en général. Par contre, on peut s’intéresser aux déplacements instantanés des joints du problème linéarisé. Un déplacement infinitésimal d’un vecteur µ(v) àchaque sommet v ∈ V (µ(v) peut être vu comme la vitesse instantanée du joint) tel que pour chaque arête (i.e. barre) les déplacements instantanés le long de cette arête induits par ce déplacement infinitésimal aux deux extrémités sont identiques, ce qui signifie

(p(u) − p(v))(µ(u) − µ(v))=0 pourtout uv ∈ E. (1.0.1)

(Voir Figure 3.2.) Une charpente est dite infinitésimalement rigide si tous ses déplacements infinitésimaux peuvent être obtenus àpartir des vecteurs de vitesse instantanée de la restriction d’un déplacement congruent de tout l’espace sur V .

8 Chapter 1. Introduction

En fait, la rigiditéinfinitésimale est une propriétéplus forte que la rigiditélocale, mais c’est une bonne alternative àla rigiditélocale car il s’avère qu’elles co¨ıncident dans la plupart des cas, àsavoir pour les charpentes en position générique. La rigiditéinfinitésimale offre plus de prise que la rigiditélocale puisque elle peut être déterminée par le calcul du rang de la matrice dérivée du système linéaire (1.1.1) pour µ. L’étude de la rigiditéa trouvédes applications pour prédire la flexibilitéde protéines. D’autres notions de rigiditéméritent notre attention. La première se préoccupe de l’unicitéglobale des charpentes, àsavoir, si l’ensemble des contraintes de distance imposées aux barres suffit àdéterminer la charpente modulo une congruence. L’étude de la rigiditéglobale a des applications dans la localisation dans des réseaux de capteurs. Bien que la rigiditéglobale apparaˆıttrès différente de la rigiditélocale, elles sont en fait étroitement liées. Il est facile de voir que la rigiditéglobale implique la rigiditélocale. Dans l’autre direction, il est démontréqu’une charpente planaire générique est globalement rigide si et seulement si il faut enlever au moins 3 sommets pour la déconnecter et elle reste localement rigide après la suppression d’une arête quelconque. En pratique, pour calculer la position des capteurs, on peut utiliser des algorithmes basés sur la programmation semi-définie. Cependant, même lorsque la position des capteurs est unique dans l’espace (plan ou en dimension 3), ces algorithmes peuvent trouver une configuration en dimension supérieure. Pour éviter cet effet indésirable, la rigidité universelle, une notion plus forte que la rigiditéglobale est requise. Une charpente est universellement rigide si elle est exclusivement déterminée modulo une congruence dans tous les espaces. Caractériser combinatoirement la rigiditéuniverselle des charpentes génériques semble plus difficile que la rigiditéglobale, en partie car elle ne dépend pas uniquement du graphe sous-jacent, même lorsque les sommets sont dans une position générique. En fait, même en dimension 1, aucune caractérisation combinatoire de la rigiditéuniverselle est connue.

9 1.1. Frameworks and rigidity

1.1 Frameworks and rigidity

Rigidity theory originally concerns with the rigidity/flexibility of frameworks. A framework is a mathematical model describing a real structure. An early source of structures that provoked the study of rigidity is from architectural engineering. Consider a planar structure constituting of solid bars that are joined at extremities, called joints, by junctions that allow two joined bars to move freely in the plane about this joint. A natural question is whether the shape of the structure can be deformed. For example, the rectangular planar structure in Figure 1.1 can be deformed continuously to a non-rectangular one as shown by the dashed figure. We say that it is flexible. Meanwhile, the planar structure in Figure 1.2 can not be deformed continuously to another shape without being ripped apart. We say that it is rigid, or more precisely, locally rigid. •• ••

• • • • Figure 1.1: A ﬂexible bar-and-joint structure in the plane.

••

• •

Figure 1.2: A rigid bar-and-joint structure in the plane.

Such a planar bar-and-joint structure can be modeled by a pair (G, p) of a graph G = (V,E) and a map p : V → R2. The vertices V of G correspond to the joints and the edges E of G correspond to the bars. The map p determines the location of the joints. In the same way, a bar-and-joint structure in 3-space (and more generally in dimension d) can be modeled with a map p : V → R3 (and p : V → Rd, respectively). We call this pair a bar-joint framework. A continuous motion of a framework (G, p) in Rd is a continuous motion of the vertices of G in Rd which preserves the edge lengths. A framework is said to be locally rigid if every continuous motion of (G, p) preserves also the pairwise distance between all

10 Chapter 1. Introduction

• • • • • • • • • • • • • • • • • • a) b)

Figure 1.3: a) A molecule structure with atoms and bonds. b) The corresponding bar-joint framework. vertices of G. Equivalently, every continuous motion of (G, p) is the restriction of a congruent motion of the whole space. The problem of determining the local rigidity of a framework is hard in general. One way to deal with this is to consider the linearized problem where we focus on the instantaneous motion of the joints. An inﬁnitesimal motion of a framework is an assignment of a vector µ(v) to each vertex v ∈ V (µ(v) can be viewed as the instantaneous velocity of the joint) such that for each edge (i.e., bar) the instantaneous displacements along this edge induced by this inﬁnitesimal motion at two ends are the same, which means

(p(u) − p(v))(µ(u) − µ(v)) = 0 for all uv ∈ E. (1.1.1)

(See Figure 3.2.) A framework is said to be infinitesimally rigid if all its infinitesimal motions can be obtained as the instantaneous velocity vectors of a restriction of a congruent motion of the whole space on V . As a matter of fact, infinitesimal rigidity is a stronger property than local rigidity, but it is a good alternative for local rigidity since it turns out that they coincide in “most” cases, namely for frameworks in generic position. Infinitesimal rigidity is more tractable than local rigidity since it can be decided by calculating the rank of a matrix derived from the linear system (1.1.1) for µ. The study of local rigidity/flexibility has found important applications in predicting flexibility of protein molecules. We can regard a molecular structure with covalent bonds as a bar-joint framework in 3-space: each atom is a joint and each covalent bond works as a bar that fixes the distance between two atoms. Since the angles between convalent bonds of an atom are also fixed, we need to add one more edge between each pair of neighbors of an atom (Figure 1.3). As protein molecules

11 1.1. Frameworks and rigidity are often very large with thousands of atoms and bonds, the linear algebra approach by calculating the rank of the rigidity matrix becomes non efficient. The study of large frameworks necessitates the combinatorial results on the underlying graphs of these frameworks which would facilitate the design of efficient algorithms. The fundamental result by Laman (1970) asserts that the local/infinitesimal rigidity of a generic framework in the plane can be discerned by a simple counting condition on vertices and edges. This counting condition is based on the following intuitive reasoning: Every motion of a point in the plane is a combination of a horizontal and a vertical motion. So we say that a point has 2 degrees of freedom. If a framework on n vertices has no edge then it has 2n degrees of freedom. Adding an edge to the framework reduces its degrees of freedom by at most 1. However, for n ≥ 2, there are always motions that can not be blocked by edges, they are congruent motions of the framework. We can count for these 3 independent motions: 2 for translations and 1 for rotation. Therefore, in an infinitesimally rigid framework without redundant edges one should expect that

1. the framework has 2n − 3 edges in total, and 2. in each subframework with n′ vertices, there are at most 2n′ − 3 edges.

The necessity of these conditions has been known since James Clerk Maxwell’s time [81]. Laman showed that, for generic frameworks, it is also sufficient. In dimension 3 and higher, up to now, no combinatorial characterization is known for generic rigidity. Nevertheless, for the special class of frameworks that describe molecular structures, a long-conjectured combinatorial characterization, known under the name of Molecular Conjecture (by Tay and Whiteley 1984), has been proven to be true. Softwares based on this characterization such as FIRST, FRODA,. . . ([67],[105]) had been developed even before the confirmation of its mathematical correctness. Let us move to other notions of rigidity. The first one concerns with the global uniqueness of frameworks, namely, if the set of distance constraints imposed by bars is sufficient to determine the framework up to congruence. As an example, let us consider the sensor network localization problem. In a sensor network, autonomous sensors collect information on enviroment condition such as tempera- tures, pressures, chemical agents,. . . and send data through the network to a center. Though not all sensors are equipped with GPS receivers, which are costly and bat- tery consuming, to localize directly their position, the distance between some pairs of sensors can be calculated (using radio signals for close sensors, for example). A

12 Chapter 1. Introduction

• •

• • • • △ •

△ △ ••

(a) (b)

Figure 1.4: (a) A wireless sensor network. Elements with GPS receivers are represented with triangles and the others with black circles. The arrow lines represent the communicability between sensors. (b) The corresponding bar-joint framework. well-designed sensor network should allow one to locate all sensors based on this information. Such a network can be converted to a bar-joint framework by putting edges between pairs of sensors whose distances are known as well as between all sensors possessing a GPS receiver (Figure 1.4). It is evident that the first condition for the localizability is that the location of each sensor is uniquely determined by the available information on the distances. It is equivalent to the global rigidity of the corresponding framework. Although global rigidity looks quite different from local rigidity, they are actually tightly related. It is easy to see that global rigidity implies local rigidity. In the other direction, it is shown that a planar generic framework is globally rigid if and only if one must remove at least 3 vertices to disconnect it and it remains locally rigid after the removal of any edge. In practice, it is also important that the location of the sensors can be computed efficiently. To this end, semidefinite programming-based algorithms are proposed. However, even when the location of the sensors is unique in the considered space (plane or 3-space), these algorithms may find a configuration in a higher dimension. To prevent this undesirable outcome, universal rigidity, a stronger notion than global rigidity is of order. A framework is universally rigid if it is uniquely determined up to congruence in any space. Characterizing combinatorially universal rigidity of generic frameworks seems more difficult than global rigidity, partly due to the fact that it does not depend uniquely on the underlying graphs, even when the vertices are in generic position (Figure 1.5). As a matter of fact, even in

13 1.1. Frameworks and rigidity

• •

• • • • •• ••

Figure 1.5: The planar generic framework on the left is universally rigid while the one on the right is not. dimension 1, no combinatorial characterization of universal rigidity is known. Driven by practical applications, many types of frameworks are extensively studied. Rigidity properties in these models are defined in the same manner as for bar-joint frameworks: local/infinitesimal rigidity for the uniqueness under continuous/infinitesimal motions, global rigidity for the uniqueness in the same dimension and universal rigidity for the uniqueness in all dimensions. We brief here some models studied in our thesis.

1. Direction-length frameworks: This is an extended model of bar-joint frameworks where beside distance constraints we have also direction constraints between vertices.

2. Body-bar frameworks and body-bar frameworks with boundaries. In a body- bar framework, bars link solid bodies and in a body-bar framework with boundaries, some bodies are linked to a ﬁxed external environment by bars or they are simply pinned down. These models can be converted to bar-joint model but are interesting in their own right as combinatorial characterizations for rigidity in these models are obtained in all dimensions.

3. Body-length-direction frameworks: These are extended models of body-bar frameworks where we have both distance and direction constraints between bodies. We also consider diﬀerent types of bodies which allow diﬀerent types of motions.

The study of these frameworks may have applications in Computer-Aided-Design or in sensor network localization.

4. Tensegrity frameworks are a rather diﬀerent model. In these frameworks we have three types of edges corresponding to three types of constraints: bars which ﬁx distances between pairs of vertices, cables which prevent distances

14 Chapter 1. Introduction

from increasing and struts which prevent distances from decreasing. The consideration of these frameworks arose naturally in architectural engineering since materials which resist well compression are not strong in withstanding tension and vice versa. Using cable and strut-like elements helps reduce the cost and mass of the structure. Recent researches also propose considering cell structure as a tensegrity structure [56].

1.2 Our contributions and organization of the thesis

Chapter 2 reviews basic notions and facts used in our thesis, especially on graphs and digraphs, combinatorial optimization tools (matroid theory, submodular function,. . . ) as well as algebra and linear algebra. Chapter 3 gives a basic formal introduction to the rigidity theory of bar-joint frameworks. The main results of this thesis appear from Chapter 4 to Chapter 8. Below, we give a summary of these chapters.

Chapter 4: Matroid approach. This chapter presents our early approach to the problem of characterizing local/inﬁnitesimal rigidity in generic frameworks. View- ing this problem as characterizing the generic rigidity matroid, we study abstract rigidity matroids, a generalization of the generic rigidity matroid. We solve an open question by Graver, Servatius and Servatius [47] on the characterization of abstract rigidity matroids in dimension 2 and extend the result to all dimensions. We also introduce the notion of 1-extendable abstract rigidity matroids and investigate the relation between these matroids and the generic rigidity matroids. These results compose an article appeared in SIDMA [85]. Furthermore, inspired by the counting condition by Laman, we propose the study of intersecting submodular functions that induce abstract rigidity matroids and obtain a characterization for these functions.

The core of our thesis lies in constructive approaches to the problem of rigidity. By “constructive approach” we first mean an approach that focuses on inductive constructions of rigid frameworks or graphs and the effect of extension operations on frameworks or graphs. This approach has proved its power particularly in characterizing infinitesimal/local rigidity (see, e.g., [101, 99, 98, 94, 72]). It is also sucessfully employed in characterizing global rigidity in [58, 24]. Second, an alternative to inductive construction is to find a decomposition for underlying graph

15 1.2. Our contributions and organization of the thesis of (minimally) rigid frameworks. Such a decomposition can be used to construct an explicit rigid framework. This approach often results in a simpler proof for the desired characterization. (See [100, 107, 60] for examples.)

Chapter 5: Inductive construction and decomposition of graphs. First, in Section 5.2, we provide an inductive construction and a decomposition for graded sparse graphs. These graphs arise when one considers frameworks with mixed constraints, then different types of edges are subject to different sparsity conditions. We also study the graded sparse matroids determined by these graphs, obtain the rank formula as well as a decomposition of these matroids. The result in this section is from a joint work with Bill Jackson [66]. Section 5.3 considers another notion of sparse graphs: (b, l)-sparse graphs, where the sparsity of subgraphs depend on their vertices. The motivation for this notion comes from the consideration of frameworks with bodies of different dimensions. We give an inductive construction for these graphs and also a characterization of these graphs as resulting graphs in a pebble game. Our results on these different types of sparse graphs generalize the classic result of Nash-Williams [83] on the decomposition of graphs into edge-disjoint spanning trees. The dual result of this is about packing of edge-disjoint spanning trees by Tutte [104] and Nash-Williams [82]. Their results can be obtained easily from the directed counterpart on packing of arc-disjoint arborescences by Edmonds [31] via an orientation result by Frank. Our third contribution in Chapter 5 is a generalization of the result of Edmonds to packing of arborescences whose roots are constrained to some matroidal condition (Section 5.4). Using a general orientation result by Frank [38], we obtain a short proof for a result of Katoh and Tanigawa which generalizes Tutte and Nash- Williams’ result for characterizing rigidity of frameworks with boundaries. This is the result of a joint work with Olivier Durand de Gevigney and Zoltán Szigeti which appeared in SIDMA [28].

Chapter 6: Infinitesimal rigidity. This chapter focuses on the infinitesimal rigidity of several types of frameworks with mixed constraints. In Section 6.3, we obtain combinatorial characterizations of infinitesimal rigidity in generic body- length-direction frameworks, using the decomposition for graded-sparse graphs in Chapter 5. This part is also from the previously mentioned joint work with Bill Jackson. Section 6.4 discusses extension operations for direction-length frameworks in general dimension. We extend the definition of 0-extension and 1-extension in R2

16 Chapter 1. Introduction to d-dimension case and investigate the effect of these extension operations on the infinitesimal rigidity and boundedness of generic frameworks. Chapter 7: Global rigidity of direction-length frameworks. We extend a result of Jackson and Jordán on the global rigidity preservingness of 1-extensions on generic direction-length frameworks in R2 to all dimensions. This chapter consists of an article appeared in IJCGA [86]. Chapter 8: Universal rigidity. While local/infinitesimal rigidity and global rigidity are well studied with abundant results, especially for generic frameworks in dimension 1 and 2, little is known about universal rigidity even in dimension 1. We offer the study of universal rigidity in two directions. First, in Section 8.2, we explore universal rigidity on the line. We obtain a complete characterization for universal rigidity of complete bipartite frameworks on the line and show that the only generically universally rigid bipartite graph in d R is the single edge K1,1 for all d ≥ 1. Many open questions and conjectures inspired by our result are discussed. This section consists of a joint article with Tibor Jordán [70]. The second direction is to relax the condition on the genericity of frameworks. In Section 8.3, we strengthen earlier results on sufficient condition for universal rigidity of bar-joint frameworks [8] and tensegrity frameworks [19] to allow config- urations in non-general position. This is the result from a joint work with Abdo Alfakih [6].

17 1.2. Our contributions and organization of the thesis

18 Chapter 2

Preliminaries

Contents 2.1 Basic notions ...... 20 2.2 Graphs and digraphs ...... 20 2.3 Matroid theory ...... 23 2.4 Submodular functions ...... 26 2.5 Algebra and linear algebra ...... 27 2.6 Congruent motions and motions of rigid bodies . . . . 30

19 2.1. Basic notions

2.1 Basic notions

We will use Z and Z+ to denote the set of integers and non negative integers, respectively. The set of rational numbers and real numbers are denoted by Q and R respectively. Rn denotes the n-dimensional Euclidean space. 0 denotes a zero matrix of appropriate dimensions or the origin of an Euclidean space. The n Euclidean norm of a vector/point x = (x1, . . . , xn) ∈ R is denote by kxk, that is

2 2 2 kxk = x1 + ··· + xn.

n The number of 2-element subsets of an n-element set is denoted by 2 . For a ﬁnite set S the number of elements of S is denoted by |S|. The collection of all the subsets of S is denoted by 2S. If a set A is a subset of a set B then we write A ⊆ B. We also write A ⊂ B when A ⊆ B and A =6 B and say that A is a proper subset of B.A multiset is a generalization of the notion of set, where each member may appear more than once. For two sets A, B, let A \ B or A − B denote the set of all elements belonging to A but not to B. A ∪ B and A ∩ B denote the union and intersection of A and B respectively. We also write A − x for A \{x} and A + x or A ∪ x for A ∪ {x}.

A partition of a non-empty set S is a collection P = {P1,...,Pn} where

P1,...,Pn are non-empty pairwise disjoint subsets of S such that P1 ∪· · ·∪Pn = S. Let f : X → Y be a map, A ⊆ X, and B ⊆ Y . The image of A by f is f(A) = {f(x): x ∈ A} and the pre-image of B by f is f −1(B) = {x ∈ A : f(x) ∈ B}. n m The Jacobian matrix of a diﬀerentiable map f : R → R , f(x1, . . . , xn) =

(f1(x1, . . . , xn), . . . , fm(x1, . . . , xn)), at a point p is the m × n matrix

∂f ∂f 1 ... 1 ∂x1 ∂xn  . . .  df|p = . .. . ∂f ∂f   m m   ...   ∂x1 ∂xn    2.2 Graphs and digraphs

A graph (or undirected graph) is a pair G = (V,E) where V is a non-empty ﬁnite set and E consists of 2-element multisubsets of V . The elements of V are called vertices or nodes of G and the elements of E are called the edges of G. Sometimes, to be more precise, we use V (G) and E(G) to refer to the vertex set and the

20 Chapter 2. Preliminaries edge set of G. An edge {u, v} of G is often denoted by uv and u, v are called the extremities or ends of the edge uv. We say that the edge uv is incident to u and v. If u = v then the edge uv is called a loop. A graph is often depicted by a set of dots for the vertices and line segments or curves for the edges. A graph without loops and multiple edges is called a simple graph. The degree of a verter v in a graph G, denoted by d(v), is the number of non-loop edges incident to v plus twice the number of loops at v. Two simple graphs G = (V,E) and G′ = (V ′,E′) are said to be isomorphic if there exists a bijection φ : V → V ′ such that for every u, v ∈ V , uv is an edge of G if and only if φ(u)φ(v) is an edge of G′. A complete graph is a simple graph where between each pair of vertices there is an edge. A complete graph on n vertices is often denoted by Kn. We also use K(V ) to denote the complete graph on a vertex set V .A bipartite graph is a graph where the vertex set can be partitioned into two sets X and Y such that there is no edge between two vertices in X and there is no edge between two vertices in Y . A complete bipartite graph is a bipartite graph with a vertex partition X,Y such that for each pair of vertices x ∈ X and y ∈ Y , xy is an edge. If |X| = m and |Y | = n, the complete bipartite graph with the vertex partition X,Y is denoted by Km,n. Let G = (V,E) be a graph and X,Y be subsets of V . Throughout this thesis, we will use E(X) to denote the set of edges induced by X in G, i.e., those edges of G with both extremities in X, and i(X) to denote |E(X)|. If X = {v}, i(X) = i(v) is simply the number of loops at v. We will use E(X,Y ) to denote the set of edges from X to Y , i.e., those edges of G with one extremity in X and the other in Y . We denote |E(X,Y )| by i(X,Y ). A subgraph of G = (V,E) is a graph H = (V ′,E′) with V ′ ⊆ V and E′ ⊆ E(V ′). It is called an induced subgraph of G if E′ = E(V ′). The vertex set of an edge set F is denoted by V (F ). A path from a vertex u to a vertex v in a graph G = (V,E) is a sequence u = v0, e1, v1, e2, . . . , vk−1, ek, vk = v of vertices v0, . . . , vk in V and edges e1, . . . , ek in E such that vi−1, vi are two ends of ei for i = 1, . . . , k. We often identify a path with its sequence of edges. The edge set may be empty if u = v. If there is a path from u to v in G then we say that v is reachable from u. A connected graph is a graph G = (V,E) whose vertex set V can not be partitioned into two nonempty subsets X and Y such that i(X,Y ) = 0. Equivalently, a connected graph is a graph G such that, for each pair of vertices u, v, there is a path from u to v in G.A connected component of a graph G is a maximal con-

21 2.2. Graphs and digraphs nected subgraph of G. Obviously, connected components of G are vertex-disjoint. An edge e of G is called a cut-edge if the deletion of e from G increases its number of connected component. A cycle is a connected graph in which each vertex has degree 2. If an edge of G is not a cut-edge then it must belong to a cycle in G.A forest is a graph that does not contain any cycles as its subgraphs. A tree is a connected forest. A subgraph H of G is said to be spanning if V (H) = V (G). A tree is a spanning tree of G if it is a spanning subgraph of G. Let k be a positive integer. A graph G is said to be k-vertex-connected, or simply k-connected, if |V (G)| > k and one has to remove at least k vertices to disconnect the graph. Note that 1-connected graphs are simply connected graphs. Let P be a partition of the vertex set V of G into non-empty subsets. We denote by EG(P) the set of all edges in E(X,Y ) for every X,Y ∈ P,X =6 Y . Edges in

EG(P) are called crossing edges of the partition P. We often denote |EG(P)| by eG(P). A directed graph (or a digraph) is a pair D = (V,A) where V is a ﬁnite set and A consists of ordered pairs of elements of V (where we can have pairs of the same element). The elements of V are called the vertices of D and the element of A are called the arcs of D. An arc (u, v) is often denoted by uv; u is called the tail and v is called the head of the arc uv. If u = v then the arc uv is called a loop. A digraph is depicted in the same way as an undirected graph except that we use an arrow to denote the direction from u to v of an arc uv. An arc uv from u to v is said to be an out-arc of u and an in-arc of v. Let D = (V,A) be a digraph and X a subset of V . An arc uv with u ∈ V \ X and v ∈ X is said to be an entering arc of X. An outgoing arc of X is an entering − arc of V \X. The set of all arcs of D entering X is denoted by RD(X), the number − of its elements is denoted by ρD(X). When X = {v}, we simply write RD(v) and + ρD(v). Concerning the outgoing arc of X we will use RD(X), δD(X) respectively. A dipath from a vertex u to a vertex v in a digraph D = (V,A) is a sequence u = v0, a1, v1, a2, . . . , vn−1, an, vn = v of vertices v0, . . . , vn in V and arcs a1, . . . , an in A such that vi−1 is the tail and vi is the head of the arc ai for i = 1, . . . , n. We often identify a dipath with the sequence of its arcs. If there exists a dipath from v to u in D then we say that u is reachable from v in D. We say that D is an r-arborescence if D is a directed tree, r is a vertex of D of in-degree 0 and all the other vertices of D are of in-degree 1. We note that an r-arborescence may consist of only the vertex r and no arc. Note also that

22 Chapter 2. Preliminaries an r-arborescence has a unique vertex of in-degree 0, namely r. We also use out- arborescence to refer to an arborescence when we want to distinguish it with an in-arborescence, which is a directed graph where one vertex has out-degree 0 and all the other vertices have out-degree 1. A sub-digraph H of D is called spanning if its vertex set V (H) coincides with V . A digraph D is strongly connected if for every two vertices u, v of D, u is reachable from v and vice versa. D is said to be k-connected if |V (D)| > k and one need to remove at least k vertices to make D non strongly connected.

2.3 Matroid theory

Definition 2.3.1 (Matroid). A matroid M is a pair (S, I) of a finite set S and a collection I of subsets of S that satisfies the following independence axioms.

(I0) ∅ ∈ I. (I1) If I ∈ I and J ⊆ I then J ∈ I. (I2) If I,J ∈ I and |I| > |J| then there exists x ∈ I \ J such that J ∪ x ∈ I.

Matroid is a structure that generalizes the notion of linear independence. If S is a finite set of vectors in a vector space, the collection I of linearly independent subsets of S verifies the three properties above. We call this matroid the linear matroid defined by S. Matroid theory draws also motivations from graph theory. Given a graph G = (V,E), the edge sets of all subgraphs of G that are forests form the independent sets of a matroid on E. We call this matroid the graphic matroid of G. In fact, this matroid is also a linear matroid. The set S is called the ground set of the matroid M and the sets in I are called the independent sets. Sets that are not independent are said to be dependent. Minimal dependent sets are called circuits. If {x} is a circuit then x is called a loop. Two elements x, y of a matroid M are said to be parallel if {x, y} is a circuit of M. A base of a matroid M is a maximal inclusionwise independent set. The collection B of bases of a matroid M satisfies the following properties.

(B0) B is not empty.

(B1) If B1 and B2 are in B then |B1| = |B2|.

23 2.3. Matroid theory

(B2) (Exchange axiom) If B1,B2 ∈ B and x ∈ B1 \ B2 then there exists

y ∈ B2 \ B1 such that B1 − x + y ∈ B.

These axioms on bases can be used to define a matroid. In fact, if a collection B of subsets of a finite set S verifies the axioms (B0), (B1), (B2) then it determines uniquely a matroid by defining the independent sets as all the subsets of bases. Furthermore, if we set B∗ = {S \ B : B ∈ B} then B∗ also satisfies (B0), (B1), (B2) and therefore determines a matroid on S. We call this matroid the dual matroid of M. Bases and circuits of the dual matroid of M are called cobases and cocircuits of M. The complement of a cocircuit is called a hyperplane of M. Let M = (S, I) be a matroid. The rank of a set X ⊆ S is the maximum cardinality of an independent subset of X:

rM(X) = max{|I| : I ⊆ X,I ∈ I}.

When the matroid is clear from the context, we may write r(X) instead of rM(X). The rank of a matroid satisﬁes the following properties.

(R1) 0 ≤ r(X) ≤ |X| for every X ⊆ S. (R2) If X ⊆ Y then r(X) ≤ r(Y ) for every X,Y ⊆ S. (R3) (Submodularity) If X,Y ⊆ S then r(X) + r(Y ) ≥ r(X ∪ Y ) + r(X ∩ Y ).

A matroid is also determined uniquely by its rank function, i.e, if r : 2S → Z+ satisﬁes the three axioms (R1), (R2) and (R3) then it deﬁnes a matroid on S: the independent sets are subsets X of S with r(X) = |X|.

The closure operator clM(·) of a matroid M on S is an operator on the collection of subsets of S deﬁned by

clM(X) = {x ∈ S : r(X ∪ x) = r(X)}. for every X ⊆ S. We may write cl(X) when M is clear from the context. Given a matroid M = (S, I) on a ﬁnite set S and a subset S′ of S, we can obtain a matroid on S′ by deﬁning

I′ = {I ∈ I : I ⊆ S′}.

24 Chapter 2. Preliminaries

It is easy to see that I′ = {I ∩ S′ : I ∈ I}. The fact that I′ veriﬁes (I0), (I1) and (I2) follows from that of I. Hence, I′ forms the independent sets of a matroid on S′. We call this matroid M′ = (S, I′) the restriction of M on S′. A free matroid on a ﬁnite set S is the matroid on S whose independent sets are all the subsets of S.

Count matroids and sparse graphs

Let us consider a class of matroids that plays an important role in combinarial rigidity theory: count matroids.

Let G = (V,E) be a graph and b : V → Z+. Here we use the bold symbol to emphasize the fact that b is a map. For the image of v ∈ V and X ⊆ V we write b(v), b(X) respectively. Let bmin denote min{b(v): v ∈ V }. Suppose that l is an integer with 0 ≤ l < 2bmin for every uv ∈ E. Then it is not diﬃcult to show that the collection

I = {F ⊆ E : iF (X) ≤ b(X) − l for every X ⊆ V with iF (X) > 0} where iF (X) denotes the number of edges in F with both extremities belonging to X, forms the independent sets of a matroid on E (see, e.g., [39, Theorem 13.5.1] for a proof). Note that, by our assumption 0 ≤ l < 2bmin, the condition for the independence of an edge set F implies that if |X| ≥ 2 then iF (X) ≤ b(X) − l. The matroid (E, I) is called a (b, l)-count matroid on G. If b(v) = 1 for all v ∈ V and l = 1, the (b, l)-count matroid on G coincides with the graphic matroid on G. When b(v) = 2 for all v ∈ V and l = 3, the (b, l)-count matroid coincides with the 2-dimensional generic rigidity matroid (more details in Chapter 3). A graph G with the edge set independent in a (b, l)-count matroid on G is called a (b, l)-sparse graph.A(b, l)-tight graph is a (b, l)-sparse graph G = (V,E) with |E| = b(V ) − l. When all b(v) take the same value m, we also refer to these graphs as (m, l)-sparse graphs and (m, l)-tight graphs, respectively. In some context, to emphasize the sparseness, we also say tight sparse graphs. Familiar examples are (1, 1)-sparse graphs which are actually forests, and (1, 1)-tight graphs which are trees.

Matroid union

One way to get a matroid from other matroids is taking their unions. Let M1 =

(S1, I1),..., Mk = (Sk, Ik) be k matroids. The matroid union of M1,..., Mk is

25 2.4. Submodular functions

the matroid M = (S, I) where S = S1 ∪ · · · ∪ Sk and

I = {I1 ∪ · · · ∪ Ik : I1 ∈ I1,...,Ik ∈ Ik}.

If Si =6 S, we can extend the matroid Mi to a matroid on S by deﬁning all elements of S \ Si to be loops. We identify this new matroid with Mi. Therefore, for the sake of convenience, we may suppose that Mi are matroids on the same ground set S. Many fundamental results become clear when viewed under the matroid union angle. For instance, the result of Nash-Williams that (k, k)-sparse graphs are the union of k edge-disjoint forests can be stated as every (k, k)-sparse matroid is the union of k (1, 1)-sparse matroids (more details in Chapter 5). The rank function for the matroid union is determined as in the following result. (For a proof, see e.g. [87, Theorem 12.3.1 ].)

Theorem 2.3.1. Let M be the matroid union of M1,..., Mk with rank functions rM1 , . . . , rMk . Then the rank function of M is given by

rM(X) = min{rM1 (Y ) + ··· + rMk (Y ) + |X \ Y | : Y ⊆ X} (2.3.1) for every X ⊆ S.

2.4 Submodular functions

Definition 2.4.1 (Submodular function). Let S be a finite set. An integer-valued function f : 2S → Z is said to be submodular if it satisfies

f(X) + f(Y ) ≥ f(X ∪ Y ) + f(X ∩ Y ) (2.4.1) for every subset X,Y of S.

If f : 2S → Z satisﬁes inequality (2.4.1) for every X,Y ⊆ S with X ∩ Y =6 ∅ then it is called an intersecting submodular function. A function g : 2S → Z is a supermodular function (intersecting supermodular function, resp.) if −g is submodular (intersecting submodular, resp.). Submodular functions (and hence supermodular functions) play an important role in combinatorial optimization because of its omnipresence. For example in a graph G, ρ(X) is a submodular function, i(X) is a supermodular function, for X ⊆ V (G), and m|V (F )|−l is an intersecting submodular function, for F ⊆ E(G), where 1 ≤ m < 2l. Moreover, submodular functions are of special interest in combinatorial optimization as they can be minimized in polynomial time.

26 Chapter 2. Preliminaries

Theorem 2.4.1 (Iwata, Fleischer and Fujishige [57], Schrijver [92]). Given a submodular function f : 2S → Z, a set U ⊆ S that minimizes f(U) can be found in polynomial time.

Submodular functions are closely related to matroids. As mentioned above, the rank of a matroid is a submodular function. Conversely, given a nondecreasing intersecting submodular function f : 2S → Z, let us consider the collection

I(f) = {I ⊆ S : |J| ≤ f(J) for every J ⊆ I,J =6 ∅}.

Theorem 2.4.2. I(f) forms the independent sets of a matroid M(f) on S. The rank function of M(f) is determined by

rM(f)(X) = min{|X0| + f(Xi): {X0,X1,...,Xt} partitions X} (2.4.2) i=1 X for X ⊆ S.

(For a proof, see [39], for example.) The circuits of this matroid are minimal non empty sets C such that f(C) < |C| and f(C) ≥ f(C − x) ≥ |C| − 1 for every x in C. Therefore, we have the following.

Proposition 2.4.3. If C is a circuit of the matroid M(f) induced by a nondecreasing intersecting submodular function f then f(C) = f(C − x) = |C| − 1.

2.5 Algebra and linear algebra

Let K be a subfield of the complex field C. By K[X1,...,Xn] we denote the ring of all n-variable polynomials f[X1,...,Xn] with coefficients in K.

The extension field of a field K ⊆ C by p1, . . . , pn ∈ C, denoted by K(p1, . . . , pn), is the smallest subfield of C that contains p1, . . . , pn. Equivalently, K(p1, . . . , pn) =

{f(p1, . . . , pn)/g(p1, . . . , pn): f, g ∈ K[X1,...,Xn], g(p1, . . . , pn) =6 0}. The algebraic closure of a ﬁeld K ⊆ C is the smallest ﬁeld K ⊆ C such that every polynomial f(X1) ∈ K[X1] has a root in K.

A set {p1, . . . , pn} ⊂ C is said to be algebraically independent over K if there does not exist a polynominal f(X1,...,Xn) ∈ K[X1,...,Xn], non identical to zero, such that f(p1, . . . , pn) = 0. It turns out that this algebraic independence is in fact matroidal.

27 2.5. Algebra and linear algebra

Theorem 2.5.1 ([87] Theorem 6.7.1). Let L be an extension field of a field K and S a finite subset of L. Then the collection I of subsets of S that are algebraically independent over K is the set of the independent sets of a matroid on S.

For an m × n matrix M we often use Mij to denote the (i, j)-entry of M. The T T transpose of M is the n × m matrix denoted by M such that Mij = Mji.

We use In to denote the identity matrix of order n, that is the n × n matrix with all (i, i)-entries being 1 and all the other entries being 0. When the dimension n is clear from the context we simply write I for In. An n × n real matrix M is T T said to be orthogonal if MM = M M = In. Sometimes we deal with matrices M whose entries are functions of a parameter d t. We denote such a matrix by M(t). The derivation M(t) is simply the matrix dt d whose (i, j)-entries are M(t) . It is routine to verify that dt ij d d d M(t)N(t) = M(t) N(t) + M(t) N(t). dt dt dt The nullspace Ker M of a matrix M with n columns is the linear space of all vector x ∈ Rn such that Mx = 0. For an n×n matrix M, let trace (M) denote the sum of all the diagonal entries of M. Let A be an n × m matrix and B an m × n matrix, then it is easy to verify that

trace (AB) = trace (BA).

An n × n-matrix A = (aij) is symmetric if aij = aji for all 1 ≤ i, j ≤ n. The set of all real n × n matrices is denoted by Sn. Then Sn is a vector space and we can deﬁne an inner product h· , ·i in Sn by

hA,Bi = trace (AB), for A, B ∈ Sn. Let M be an n × n real matrix. A scalar λ is said to be an eigenvalue of M if there exists x ∈ Rn, x =6 0 such that Mx = λx. A symmetric n × n matrix A is said to be positive definite if for every non-zero x ∈ Rn we have xT Ax > 0. A symmetric n×n matrix A is positive semidefinite (or PSD for short) if for every x ∈ Rn the inequality xT Ax ≥ 0 holds. PSD matrices are used to characterize the universal rigidity of frameworks (see Chapter 8). A principal minor of an n × n matrix A is the determinant of a square submatrix of A with rows and columns indexed by the same subset X of {1, . . . , n}. If X = {1, . . . , k} for some k ≤ n then the principal minor is called a leading principal minor. The following lemma summarizes equivalent statements about the positive semidefiniteness of a matrix.

28 Chapter 2. Preliminaries

Lemma 2.5.2. Let A ∈ Sn. The following statements are equivalent.

1. A is a PSD matrix.

2. All eigenvalues of A are non-negative.

3. All principal minors of A are non-negative.

4. All leading principal minors of A are non-negative.

5. A = XXT for some n × n matrix X.

A consequence of Lemma 2.5.2 is the following.

Lemma 2.5.3.

1. If A is a PSD matrix in Sn then trace (A) ≥ 0 and trace (A) = 0 if and only if A = 0.

2. Let A, B be two PSD matrices in Sn, then hA,Bi ≥ 0 and hA,Bi = 0 if and only if AB = 0.

Proof. 1. By Lemma 2.5.2 we may assume that A = XXT for some n × n matrix

X = (xij). Then

T 2 trace (A) = Xij(X )ij = xij ≥ 0. i,j i,j X X

Moreover, trace (A) = 0 if and only if xij = 0 for all i, j, which means that X = 0 and hence A = 0.

2. We may suppose that A = XXT and B = YY T for some n × n matrices X,Y . Then

hA,Bi = trace (AB) = trace (XXT YY T ) = trace (Y T XXT Y ) = trace ((XT Y )T (XT Y ) ≥ 0.

Moreover, trace (AB) = 0 if and only if the trace of the PSD matrix (XT Y )T (XT Y ) is 0, which implies XT Y = 0 by the ﬁrst statement and hence AB = 0 holds.

A n × n matrix A = (aij) is skew-symmetric if aij = −aji for all 1 ≤ i, j ≤ n; in particular, aii = 0 for all 1 ≤ i ≤ n. Skew-symmetric matrices will be used to describe inﬁnitesimal rotations of rigid bodies.

29 2.6. Congruent motions and motions of rigid bodies

d Let x, y be two vectors in R . The join of x = (x1, . . . , xd) and y = (y1, . . . , yd) d is the 2 -dimensional vector (1,2) (1,3) (i,j)(d−1,d)

x1 x2 x1 x3 xi xj xd−1 xd x ∨ y = , ,..., ,..., y1 y2 y1 y3 yi yj yd−1 yd !

d Let A = (a ij) be a d × d skew-symmetric matrix. We deﬁne a 2 -dimensional vector w associated with A by w = (a12, a13, . . . , a(d−1)d), i.e., by putting aij consecutively in the lexicographic order. A related notion to join in R3 is the cross product

x x x x x x x × y = 2 3 , 3 1 , 1 2 y2 y3 y3 y1 y1 y2 !

which diﬀers from x ∨ y in the order and a possible −1 scaling of entries. The following equality relates inner product and join.

hAy , xi = hw , x ∨ yi. (2.5.1)

Indeed,

hAy , xi = aijyjxi i,j X = aijyjxi + aijyjxi (since aii = 0 for every i) ij X X = wijyjxi − wijyixj i

2.6 Congruent motions and motions of rigid bodies

A congruence of Rd is a map h : Rd → Rd such that

kh(p) − h(q)k = kp − qk for all p, q ∈ Rd. (2.6.1)

The following result is fundamental, a proof can be found in [77, 22].

30 Chapter 2. Preliminaries

Proposition 2.6.1. A map h : Rd → Rd is a congruence if and only if there exists an orthogonal d × d matrix M such that

h(p) = Mp + h(0) for all p ∈ Rd.

A congruent motion of Rd is a map P : [0, 1] × Rd → Rd such that P (t, p) is continuously diﬀerentiable in p and t and for each t ∈ [0, 1], the map P (t, ·): Rd → Rd is a congruence, i.e.,

P (t, p) = M(t)p + P (t, 0) for all t ∈ [0, 1] and p ∈ Rd, (2.6.2) where M(t) is a d × d orthogonal matrix. d Let A(t) = M(t), differentiating both sides of the equality M(t)M T (t) = I dt we have, M(t)A(t)T + A(t)M(t)T = 0. At t = 0, M(0) = I, so we obtain A(0)T + A(0) = 0, i.e., A = A(0) is a skew- symmetric matrix. By 2.6.2, the infinitesimal motion of p induced by the congruent motion P , i.e., the instantaneous velocity of p at instant t = 0, is given by d P (t, p)| = Ap + t. dt t=0 d Here t = P (t, 0)| which corresponds to the instantaneous translation, while dt t=0 A corresponds to the instantaneous rotation. Inversely, suppose that A is a d × d skew-symmetric matrix and t ∈ Rd. Let t2 tn M(t) = I + tA + A2 + ··· + An + · · · ≡ etA. 2! n! Then it is easy to show that this series converges for every t ∈ R, so M(t) is well-defined. Moreover, M(t) is infinitely differentiable and

d M(t)| = A. dt t=0

T T On the other hand, M(t)M(t)T = etAetA = etA+tA = I, since A is skew- symmetric. Hence M(t) is orthogonal and

P (t, p) = M(t)p + tt for all t ∈ [0, 1] and p ∈ Rd, is a congruent motion whose inﬁnitesimal motion is Ap + t for all p ∈ Rd.

31 2.6. Congruent motions and motions of rigid bodies

Therefore, an infinitesimal motion of a congruent motion (or infinitesimal congruence for short) of Rd can be described by a pair (A, t) of a skew-symmetric d×d matrix A and a vector t ∈ Rd. In particular, the space of infinitesimal congruences d d+1 of R is of dimension 2 . A rigid body is an idealization of a solid body in physical world. In a rigid body, the distance between any two points is constant. When there is no specification, a rigid body B in Rd, or simply a body, is understood to be of full dimension, i.e, the set of points of B affinely spans Rd A motion of a rigid body B ⊂ Rd is a map P : [0, 1] × B → Rd such that, for every p ∈ B, the map P (·, p) is continuous and

kP (t, p) − P (t, q)k = kP (0, p) − P (0, q)k, for every t ∈ [0, 1], (2.6.3) that is, the motion preserves the distance between points in the rigid body B, or equivalently speaking, P (t, ·) is an isometry of B for every t ∈ [0, 1]. In fact, P (t, p) is the position of a point p in B at time t. These isometries P (t, ·) can be extended uniquely to congruences of Rd [47, 22]

P (t, p) = M(t)p + P (t, 0).

The above discussion shows that the inﬁnitesimal motion of a rigid body in Rd can also be described by a pair (A, t) of a skew-symmetric matrix A and a vector t ∈ Rd.

32 Chapter 3

Rigidity theory

Contents 3.1 Introduction ...... 34 3.2 Various types of rigidity ...... 35 3.2.1 Local rigidity ...... 35 3.2.2 Infinitesimal rigidity and rigidity matrix ...... 36 3.2.3 Local rigidity versus infinitesimal rigidity ...... 38 3.2.4 Static rigidity ...... 40 3.2.5 Infinitesimal rigidity versus static rigidity ...... 41 3.2.6 Global rigidity ...... 41 3.2.7 Universal rigidity ...... 42 3.2.8 Stress matrices ...... 42 3.3 Combinatorial rigidity ...... 44

33 3.1. Introduction

3.1 Introduction

The theory of rigidity is characterized by its diversity from many aspects: motivations and applications, techniques, and, especially, models. It is thus difficult to cover the basic concepts in all prevailing models in this modest introduction into rigidity theory. In this chapter, we content ourselves with a description of the theory of rigidity for the bar-joint model. The first reason is that this model is easy to describe. Secondly, many other models can be converted to this model. Thirdly, ideas and techniques from the study of this model can be applied to other models as well. Lastly, although being quite simple to describe, the problems of characterizing the rigidity in the bar-joint model remains among the most difficult problems in rigidity theory. Therefore, bar-joint model serves as a base model to understand basic concepts, techniques as well as challenges in rigidity theory. A d-dimensional bar-joint framework is a pair (G, p) of a simple graph G = (V,E) and an embedding p which maps each vertex v ∈ V to a point p(v) in Rd. The vertices and the edges of G model the joints and the bars of a bar-and-joint structure while p is the placement of the structure in the d-dimensional Euclidean space. The pair (G, p) is also called a d-dimensional bar-joint realization of G and sometimes we refer to p as a configuration of V in Rd. The affine dimension of a configuration p is the dimension of the affine space spanned by {p(v): v ∈ V }. We say that a configuration p or a framework (G, p) in Rd is of full dimension if the affine dimension of p is d. The embedding p can also be regarded as a point p ∈ Rd|V |. Two realizations (G, p) and (G, q) of G in Rd are said to be equivalent if

kp(u) − p(v)k = kq(u) − q(v)k, for every edge uv in E, i.e., the length of the bars are the same in these two realizations. They are said to be congruent if

kp(u) − p(v)k = kq(u) − q(v)k, for every pair of vertices u, v in V.

Deﬁnition 3.1.1 (Rigidity map). The d-dimensional rigidity map of a graph G is d|V | |E| the map fG : R → R deﬁned by

uv∈E 2 T fG(p) = (..., kp(u) − p(v)k ,... )

Note that two frameworks (G, p) and (G, q) are equivalent if and only if fG(p) = fG(q). They are congruent if fK(V )(p) = fK(V )(q).

34 Chapter 3. Rigidity theory

3.2 Various types of rigidity

This section introduces basic concepts about various types of rigidity and discusses the relation between these types.

3.2.1 Local rigidity

The most visual way to talk about local rigidity is to use continuous deformations as in Chapter 1. We formalize this deﬁnition as follows. A continuous motion from a framework (G, p) to a framework (G, q) in dimension d is a map

P : [0, 1] × V → Rd such that

1. P (0, v) = p(v) for all v ∈ V , 2. P (1, v) = q(v) for all v ∈ V , 3. P (·, v) is continuous for all v ∈ V , 4. kP (t, u) − P (t, v)k = kp(u) − p(v)k for all uv ∈ E(G) and t ∈ [0, 1].

We say that P is a smooth motion if P is a continuous motion and P (·, v) is infinitely differentiable for each v ∈ V . Gluck [43] showed that the following three definitions of local rigidity are equivalent.

Deﬁnition 3.2.1 (Local rigidity–continuous). A framework (G, p) in Rd is said to be locally rigid if every continuous motion of (G, p) in Rd results in a framework (G, q) that is congruent to (G, p).

Deﬁnition 3.2.2 (Local rigidity–topological). A framework (G, p) in Rd is locally nd rigid if there exists a neighborhood Np ⊂ R of p such that for every equivalent realization (G, q) with q ∈ Np we must have that (G, q) is congruent to (G, p).

Deﬁnition 3.2.3 (Local rigidity–analytic). A framework (G, p) in Rd is locally rigid if every smooth motion of (G, p) in Rd results in a framework (G, q) that is congruent to (G, p).

The topological deﬁnition above gave rise to the term “local rigidity”.

Deﬁnition 3.2.4 (Flexible). A framework is called ﬂexible in Rd if it is not locally rigid in Rd.

35 3.2. Various types of rigidity

Using the rigidity map, we can also restate the topological deﬁnition of local rigidity as follows.

A d-dimensional framework (G, p) is locally rigid if for a small enough −1 −1 neighborhood Np of p, fG (fG(p)) ∩ Np = fK(V )(fK(V )(p)) ∩ Np.

3.2.2 Inﬁnitesimal rigidity and rigidity matrix

The problem of determining the rigidity of a framework becomes more tractable if we linearize the problem. Suppose that P is a smooth motion of (G, p). Then

kP (t, u) − P (t, v)k2 = kp(u) − p(v)k2 for all uv ∈ E(G) and t ∈ [0, 1].

′ Diﬀerentiating this equation at t = 0 and setting µ(u) = P (t, u)|t=0, µ(v) = ′ P (t, v)|t=0, we have

(p(u) − p(v))(µ(u) − µ(v)) = 0, for all uv ∈ E(G).

Here, µ(u) can be regarded as the instantaneous velocity of the joint u at time t = 0. This motivates us to deﬁne the inﬁnitesimal motions of a framework as follows.

Definition 3.2.5 (Infinitesimal motions). An infinitesimal motion of a d-dimensional framework (G, p) is an assignment, to each vertex v ∈ V , of a vector µ(v) ∈ Rd such that (p(u) − p(v))(µ(u) − µ(v)) = 0, for all uv ∈ E(G).

We can view an infinitesimal motion as a vector µ = (. . . , µ(u),... )T ∈ Rd|V |. Then it is not difficult to show that the infinitesimal motions of a framework (G, p) form a vector subspace of Rd|V |. Among the smooth motions of a framework, there are those arising from the congruent motions of space. The infinitesimal motions induced by these motions are called trivial infinitesimal motions. Again, these trivial infinitesimal motions form a vector subspace of the infinitesimal motion subspace.

Definition 3.2.6 (Infinitesimal rigidity). A framework (G, p) is infinitesimally rigid if its only infinitesimal motions are trivial ones. Otherwise, it is said to be infinitesimally flexible.

When the aﬃne dimension of {p(v): v ∈ V } is d, the dimension of the trivial d+1 inﬁnitesimal motion space is 2 (c.f. Section 2.6). Roughly speaking, there are 36 Chapter 3. Rigidity theory

d d independent infinitesimal motions induced by translations and 2 independent infinitesimal motions induced by rotations. When this affine dimension is strictly less than d, i.e., when all the vertices lie in a hyperplane of Rd, all infinitesimal motions are trivial if and only if G is a complete graph on at most d vertices [47]. Given a framework (G, p) in Rd, the question whether (G, p) is infinitesimally rigid can be answered by considering the rank of the so-called rigidity matrix.

Deﬁnition 3.2.7 (Rigidity matrix). The rigidity matrix R(G, p) of a d-dimensional framework (G, p) is a |E| × d|V | matrix whose rows are indexed by the edges of G and whose columns are indexed by the vertices of G such that

• each edge indexes one row; • each vertex indexes d columns; • the submatrix indexed by an edge e = uv and the vertex u and v are p(u)−p(v) and p(v) − p(u) respectively; • elsewhere all entries are zero.

That is, R(G, p) is written as

u v . . . . e=uv ··· 0 ··· p(u) − p(v) ··· 0 ··· p(v) − p(u) ··· 0 ···  . . . .     For example, the rigidity matrix of the framework in Figure 3.1 is

1 2 3 4

{1,2} 0 −1 0 1 0 0 0 0 {1,3} −1 −1 0 0 1 1 0 0  {1,4} −1 0 0 0 0 0 1 0     {2,3} 0 0 −1 0 1 0 0 0    {3,4} 0 0 0 0 0 1 0 −1      It is easy to see that the rigidity matrix is in fact half of the Jacobian of the rigidity map, i.e, 1 R(G, p) = df | . 2 G p It is also immediate from the deﬁnition that µ ∈ Rd|V | is an inﬁnitesimal motion of (G, p) if and only if µ is in the nullspace of R(G, p). Moreover, we have the following.

37 3.2. Various types of rigidity

p(2)=(0,1) p(3)=(1,1) • •

• • p(1)=(0,0) p(4)=(1,0)

Figure 3.1: A planar framework.

Proposition 3.2.1 ([9]). Let (G, p) be a framework on n vertices in Rd with |V | ≥ 2. Then the rank of R(G, p) is at most S(n, d) where

dn − d+1 , n ≥ d + 1; S(n, d) = 2 n , n ≤ d. ( 2 Moreover, a framework (G, p) on n vertices in Rd is inﬁnitesimally rigid if and only if rank R(G, p) = S(n, d).

It is worth remarking that S(n, d) is also the upper bound for the rank of the rigidity matrix of a complete framework on n vertices.

3.2.3 Local rigidity versus inﬁnitesimal rigidity

The inﬁnitesimal rigidity is in fact a stronger property than local rigidity.

Theorem 3.2.2. If a framework (G, p) is inﬁnitesimally rigid then it is locally rigid.

Several different proofs for this well-known fact can be found in [22] for example. Here we brief a proof idea by Alexandrov and Gluck [43]. If G has at most d vertices then by Proposition 3.2.1,(G, p) is infinitesimally rigid if and only if G is a complete graph, so obviously (G, p) is locally rigid. So suppose that G has at least d + 1 vertices. The fact that (G, p) is infinitesimally rigid implies that p is a regular point of fG and of fK(V ) as well. Therefore, for a small −1 enough neighborhood Np of p, fG (fG(p)) ∩ Np is a manifold whose co-dimension is rank R(G, p). But rank R(G, p) = rank R(K(V ), p) is also the co-dimension of −1 the manifold fK(V )(fK(V )(p))∩Np. This latter manifold is obviously a submanifold of the former one hence the equality between their co-dimension implies that they coincide. Therefore, (G, p) is locally rigid. The converse of Theorem 3.2.2 is not always true. Figure 3.2(a) illustrates a 2-dimensional framework which is locally rigid but not infinitesimally rigid.A

38 Chapter 3. Rigidity theory slightly different embedding of the same graph as in Figure 3.2(b) is however both locally rigid and infinitesimally rigid. In fact, the first embedding is “special” in some sense: the three vertices b, e, c are collinear. It is not the only “special embedding” that may cause the difference between local rigidity and infinitesimal rigidity. Figure 3.2(c) shows a 2-dimensional framework without three collinear vertices which is locally rigid but not infinitesimally rigid. Considering only generic embeddings helps us avoiding this unconvenient situation.

a b c •a • •• •e f• d• d• •••e •• • • b c b c a d •e (a) (b) (c)

Figure 3.2: (a) There is a nontrivial infinitesimal motion with µ(a) = µ(b) = µ(c) = µ(d) = 0 and µ(e) perpendicular to the line bec. (b) A generic embeding of the same graph as in (a). No non-trivial infinitesimal motion exists. (c) A non-generic framework where abcd is a rectangle and abe, bcf are equi- lateral right triangles. A non-trivial infinitesimal motion is depicted.

Deﬁnition 3.2.8 (Linearly generic embedding). An embedding p : V → Rd is (linearly) generic if every submatrix of R(K(V ), p) attains the maximum rank over all d-dimensional embeddings. A framework (G, p) with p being a generic embedding is called a generic framework or generic realization of G.

Theorem 3.2.3 (Asimow and Roth [9]). A generic framework is locally rigid if and only if it is inﬁnitesimally rigid.

Rigidity matroids

An important property of generic embeddings is that for two arbitrary generic embeddings and a subset F of edges, the rank of the set of rows indexed by F in the two rigidity matrices are the same. Therefore, all generic embeddings deﬁne a unique linear matroid on the edge set E of G. We call this matroid the d- dimensional generic rigidity matroid of G. This matroid can be regarded as the

39 3.2. Various types of rigidity restriction on E(G) of the d-dimensional generic rigidity matroid of the complete graph K(V ). We denote the latter by Gd(n) where n = |V |. Note that in this matroid, edge sets that induce isomorphic subgraphs have the same rank, namely, ′ this matroid depends uniquely on n and d. Again, if n ≤ n , Gd(n) can be considered ′ as a restriction of Gd(n ), so sometimes we will use Gd to mean a large generic rigidity matroid that contains all the d-dimensional generic rigidity matroids considered in our context.

3.2.4 Static rigidity

Let us consider rigidity from a structural engineering viewpoint. Suppose that at each joint v of a 3-dimensional bar-joint structure we apply a force f(v) such that the net force and the net moment (about three coordinate axes) on the whole structure is zero. Such a system F = (..., f(v),... ) of forces is called an equilibrium force. The equilibrium condition is equivalent to

f(v) = 0 and f(v) × p(v) = 0, (3.2.1) v∈V v∈V X X which is a system of 6 linearly independent equations. The ﬁrst three equations are for the three components of the net force and the last three equations are for the net moment about three coordinate axes. It follows that the space of all equilibrium forces of a 3-dimensional structure is a vector space of dimension 3|V | − 6. The structure is stable if for every equilibrium force, it can avoid the deformation by generating tensions and compressions on its bars – we say that the equilibrium force is resolved. The system (3.2.1) can be rewritten using join as

(f(v), 0) ∨ (p(v), 1) = 0. v∈V X where (f(v), 0) and (p(v), 1) are 4-dimensional vectors obtained from f(v), p(v) by adding a 0 and a 1 entry respectively. For bar-joint frameworks in general dimensions d, we say that a system of forces F = (..., f(v),... ), f(v) ∈ Rd, exerted on the joints, is in equilibrium if

(f(v), 0) ∨ (p(v), 1) = 0. v V X∈ (See [27] for more details.) Therefore, all equilibrium forces of a d-dimensional bar-joint framework of full d+1 dimension form a vector space which has dimension d|V | − 2 . 40 Chapter 3. Rigidity theory

Deﬁnition 3.2.9 (Static rigidity). A framework (G, p) is statically rigid if for every equilibrium force F = (..., f(u),... ) there is an assignment of scalar ωuv to each edge uv ∈ E such that

f(u) + ωuv(p(v) − p(u)) = 0, for every u ∈ V. (3.2.2) uv E X∈

Here, ωuv plays the role of the internal stress on the bar uv, which generates an internal force ωuv(p(u) − p(v)) at the joint u. Equation (3.2.2) means that the sum of the external force and the internal forces at each joint is zero.

3.2.5 Inﬁnitesimal rigidity versus static rigidity

Let F = (..., f(u),... ) be a system of forces in equilibrium on a framework (G, p) which is resolved by a stress ω = (. . . , ωuv,... ). Using the rigidity matrix, the system of linear equations (3.2.2) can be rewritten as

F − ωR(G, p) = 0, which means that F belongs to the row space of R(G, p). Conversely, if F = aR(G, p), where a ∈ R|E(G)| is a vector in the row space of R(G, p) then it is an equilibrium force and obviously can be resolved with ω = a. Therefore, the framework (G, p) is statically rigid if and only if the row space of R(G, p) has d+1 dimension d|V | − 2 . But it is also the necessary and sufficient condition for (G, p) to be infinitesimally rigid. Hence we obtain the equivalent between the static rigidity and the infinitesimal rigidity of bar-joint frameworks.

Theorem 3.2.4 (see, e.g., [47, 22]). A framework (G, p) is inﬁnitesimally rigid if and only if it is statically rigid.

3.2.6 Global rigidity

Deﬁnition 3.2.10 (Global rigidity). A framework (G, p) in Rd is globally rigid if for every framework (G, q) in Rd, (G, q) is equivalent to (G, p) implies that (G, q) is congruent to (G, p).

From this deﬁnition, a globally rigid framework is clearly locally rigid.

41 3.2. Various types of rigidity

3.2.7 Universal rigidity

So far, we have always considered frameworks and their motions in the same dimension. As discussed in Chapter 1, the uniqueness of the location for a localization problem, which is equivalent to the global rigidity of the associated framework, does not imply that we can find this location efficiently. The universal rigidity of a framework is a stronger property that guarantees the efficiency of the exploited SDP method.

Deﬁnition 3.2.11 (Universal rigidity). A framework (G, p) is said to be universally rigid if and only if, for every framework (G, q) in any dimension, (G, q) is equivalent to (G, p) implies that (G, q) is congruent to (G, p).

Universal rigidity as universal local rigidity

It is clear from the deﬁnitions that universal rigidity is a stronger property than global rigidity which is again stronger than local rigidity. On the other hand, the following result of Bezdek and Connelly shows that, if (G, p) and (G, q) are two realizations of G in dimension d ≤ d′ respectively, then there is a smooth motion from (G, p) to (G, q) in dimension 2d′. (We regard (G, p) and (G, q) as embedded in dimension 2d′ by adding zeros to missing coordinates.)

Lemma 3.2.5 (Leapfrog Lemma [14]). Suppose that p and q are two embeddings of G in Rm. Then the following P (t, v) is a smooth motion in R2m such that P (0, v) = p(v),P (1, v) = q(v) for v ∈ V and for 0 ≤ t ≤ 1, kP (t, u) − P (t, v)k is monotone for every u, v ∈ V : p(v) + q(v) p(v) − q(v) p(v) − q(v) P (t, v) = + (cos πt) , (sin πt) , for v ∈ V. 2 2 2 This result implies that universal rigidity can be regarded as local rigidity in a universal sense.

3.2.8 Stress matrices

Deﬁnition 3.2.12 (Self-stress). A self-stress or an equilibrium stress of a framework (G, p) is an assignment of a scalar ωuv = ωvu to each edge uv ∈ E such that for each vertex u ∈ V the following equilibrium condition is satisﬁed.

ωuv(p(u) − p(v)) = 0. (3.2.3) v∈V :uv∈E X

42 Chapter 3. Rigidity theory

p(2)=(1,0) p(3)=(1,1) • 1 • −1

1 1

−1 • • p(1)=(0,0) 1 p(4)=(1,0)

Figure 3.3: A framework with an equilibrium stress.

From the statics viewpoint discussed above, a self-stress is a stress of the bar- joint structure which resolves the zero equilibrium force. It is also immediate from the deﬁnition that ω = (ωuv : uv ∈ E) is an equilibrium stress of (G, p) if and only if ωR(G, p) = 0.

Deﬁnition 3.2.13 (Stress matrix). Let (G, p) be a framework and ω = (ωuv : uv ∈ E(G)) be an equilibrium stress of (G, p). The stress matrix of (G, p) associated with ω is the |V | × |V | symmetric matrix Ω with rows and columns indexed by vertices in V such that

−ωuv if uv ∈ E,

Ωuv =  w∈V :uw∈E ωuw if u = v,  0 otherwise.  P For example, the stress matrix associated with the stress in Figure 3.3 is

1 2 3 4

1 1 −1 1 −1 2 −1 1 −1 1 Ω =  . 3 1 −1 1 −1   4 −1 1 −1 1      Global rigidity and universal rigidity via stress matrices

It turns out that stress matrices encode rigidity properties of frameworks. Below we summarize some well-known results on the relations between stress matrices and rigidity of frameworks. These results are explained with more details in Chapter 8. First we need one more deﬁnition.

Deﬁnition 3.2.14 (Algebraically generic embedding). An embedding p : V → Rd is algebraically generic if the set of all the coordinates of p is algebraically independent over the rationals.

43 3.3. Combinatorial rigidity

Theorem 3.2.6 (Connelly [23], Gortler, Healy and Thurston [44]). An algebraically generic framework (G, p) on n ≥ d + 2 vertices in Rd is globally rigid if and only if (G, p) possesses a stress matrix of rank n − d − 1.

Theorem 3.2.7 (Connelly [21, 19], Alfakih [3], Gortler and Thurston [45]). An algebraically generic framework (G, p) on n ≥ d + 2 vertices in Rd is universally rigid if and only if (G, p) possesses a positive semideﬁnite stress matrix of rank n − d − 1.

Furthermore, Alfakih and Ye [8] shows that the “if” part of Theorem 3.2.7 still holds if one relaxes the algebraic genericity condition on the embedding to a general position condition, which requires that, under the map p, every d + 1 vertices of V are mapped to affinely independent points in Rd. This result will be strengthened in Chapter 8. For the sake of convenience, in passages concerning different types of rigidity, we will use the term “generic” both for “linearly generic” if it is about local/infinitesimal rigidity and for “algebraically generic” if it is about global/universal rigidity.

3.3 Combinatorial rigidity

One central problem of rigidity theory is that given a framework, decide whether it is rigid (locally, infinitesimally, globally or universally) or not. This problem for an arbitrary embedding is known to be untractable, in general [91], [1]. The equivalence between infinitesimal rigidity and local rigidity for generic frameworks as well as the characterization of globally and universally rigid algebraically generic frameworks through stress matrices suggest that this problem is more tractable for generic frameworks. On the other hand, it often turns out that if a rigidity property holds for a generic realization (G, p) of a graph G then it also holds for every generic realization (G, q) of G in the same dimension. If this happens, the rigidity property in question is said to be a generic property. More generally, we define:

Definition 3.3.1 (Generic property). A property P of frameworks is generic in dimension d if whenever some generic realization p of a graph G in Rd has property P, then every generic realization q of G in Rd also has property P. The above discussion about local rigidity and infinitesimal rigidity tells us that local rigidity and infinitesimal rigidity are both generic properties of bar-joint

44 Chapter 3. Rigidity theory frameworks in every dimension. More precisely, given a graph G, if there exists a d-dimensional linearly generic realization (G, p) of G that is locally (resp., infinitesimally) rigid then every d-dimensional generic realization (G, q) of G is also locally (resp., infinitesimally) rigid since the rank of R(G, p) and R(G, q) are equal. Global rigidity is also proved to be a generic property by Gortler, Healy and Thurston [44]. However, universal rigidity is not a generic property (Figure 1.5). When focusing on graphs, we say that a graph G has property P in dimension d if every generic framework of G in dimension d has property P. Note that then locally rigid, infinitesimally rigid and statically rigid mean the same thing for graphs. The study of combinatorial rigidity focuses on finding combinatorial characterizations of the underlying graphs of generic frameworks that have some rigidity property. The first fundamental result in combinatorial rigidity is the famous theorem of Laman which characterizes the underlying graphs of locally rigid generic frameworks in the plane. We refer to these graphs as generically locally rigid graphs in R2.

Theorem 3.3.1 (Laman [75]). A graph G = (V,E) is generically locally rigid in R2 if and only if E contains a subset F that satisﬁes

1. |F | = 2|V | − 3, and

2. |F ′| ≤ 2|V (F ′)| − 3 for every subset F ′ ⊆ F .

In other words, a graph is generically locally rigid in R2 if and only if it contains a (2, 3)-tight spanning subgraph. In many cases, it is convenient to consider a minimally generically locally rigid graph, i.e., a generically locally rigid graph with the property that deleting any edge makes it no more generically locally rigid. These graphs are often referred to as isostatic graphs. The following results are equivalent forms of Laman’s theorem for 2-dimensional isostatic graphs.

Theorem 3.3.2 (Lov´asz and Yemini [80], Recski [89]). For a graph G, the following statements are equivalent.

1. G is isostatic in the plane.

2. Duplicating any edge of G results in a graph that is the union of two edge- disjoint spanning trees.

45 3.3. Combinatorial rigidity

3. Adding an edge between any two vertices of G results in a graph that is the union of two edge-disjoint spanning trees.

Theorem 3.3.3 (Crapo [26]). A graph G = (V,E) is isostatic in the plane if and only if G can be decomposed into three edge-disjoint trees T1,T2,T3 such that each vertex in V is covered by exactly two of them, and no two subtrees of T1,T2,T3 with more than one vertex span the same set of vertices.

A decomposition satisfying the condition in this theorem is call a 3T 2 proper decomposition. From a matroid viewpoint, the generic local rigidity of a graph in the plane can be determined from the rank function of the generic rigidity matroid G2.

Theorem 3.3.4 (Lov´asz and Yemini [80]). The rank of G2 is given by

rG2 (E) = min{ (2|Vi| − 3) : E(V1),...E(Vm) partition E}. i=1,...,m X One important trend in rigidity theory is to study inductive construction of generically rigid graphs. Complete characterizations of generically locally rigid graphs and generically globally rigid graphs in terms of inductive construction are obtained for bar-joint model in dimension 2. (They are trivial for dimension 1.) This approach is even more fruitful in other models such as body-bar frameworks, body-bar-hinge frameworks, etc., where complete characterizations for generic local rigidity are obtained for all dimensions. Below we summarize results on generic local and global rigidity related to inductive construction. First, we recall the deﬁnition of some useful extension operations.

Deﬁnition 3.3.2 (0-extension, 1-extension).

1.A d-dimensional 0-extension on a graph H is an operation that adds to H a

new vertex v and connects v to d diﬀerent vertices v1, . . . , vd in V (H) (Figure 3.4).

2.A d-dimensional 1-extension on a graph H deletes an existing edge v1v2 in

H, adds to H a new vertex v, then connects v to v1, v2 and other d−1 vertices

v3, . . . , vd+1 in H (Figure 3.5).

A graph G obtained from a graph H by a d-dimensional 0-extension (resp., 1- extension) is called a d-dimensional 0-extension (resp., 1-extension) of H. 0- extension and 1-extension are known to preserve the isostaticity of a graph.

46 Chapter 3. Rigidity theory

v •

0-extension • • ... • • • ... • v1 v2 vd v1 v2 vd

Figure 3.4: d-dimensional 0-extension.

v •

1-extension • • ... • • • ... • v1 v2 vd+1 v1 v2 vd+1

Figure 3.5: d-dimensional 1-extension.

Theorem 3.3.5 (see [10, 102]). Suppose that H is a d-dimensional isostatic graph and G is obtained from H by a d-dimensional 0-extension or 1-extension. Then G is d-dimensional isostatic.

In dimension 2, these two operations are proved to be suﬃcient for characterizing isostatic graphs.

Theorem 3.3.6 (see, e.g., [10, 47, 58]). A graph G = (V,E) is isostatic in the plane if and only if G can be constructed from K2 by a sequence of 2-dimensional 0-extensions and 1-extensions. (See Figure 3.6.)

The problem of characterizing isostatic graphs in dimension 3 seems to be extremely diﬃcult. An example showing that the Laman-type counting condition does not work in dimension 3 is the famous double banana in Figure 3.7. It is easy to verify that, in this graph, for every subset X of V with |X| ≥ 3, i(X) ≤ 3|X|−6 and |E| = 3|V |−6. Yet the graph is not rigid in dimension 3 since there is a relative rotation of the two bananas about the axis through their two ends. A characterization of 3-dimensional isostatic graphs in terms of inductive construction may have to consider the following conjecture.

Conjecture 3.3.7 (X-replacement [102]). Let G be a 3-dimensional isostatic graph and v1, . . . , v5 distinct vertices of G with v1v2, v3v4 ∈ E(G). Then the operation that deletes v1v2, v3v4 and adds to G a new vertex v and edges vv1 . . . , vv5 results in a 3-dimensional isostatic graph.

47 3.3. Combinatorial rigidity

Figure 3.6: A sequence of graphs constructed from K2 by a sequence of 2-dimensional 0- extensions and 1-extensions. Bold edges denote new edges. All these graphs are isostatic.

Figure 3.7: The double banana.

Although a combinatorial characterization of 3-dimensional generically locally (globally) rigid graphs remains challenging, a positive result is obtained for an important class of graphs: square graphs (equivalently called molecular graphs for their role in modeling molecular structures). The square graph G2 of a graph G is obtained from G by adding an edge between every pair of vertices that have a common neighbor in G (c.f. Figure 1.3). The Molecular Conjecture, proved by Katoh and Tanigawa [72] for body-hinge structures in general dimension, implies the following combinatorial characterization for the 3-dimensional rigidity of square graphs (see [59]).

Theorem 3.3.8 (Katoh and Tanigawa [72]). The square graph G2 of a graph G is

48 Chapter 3. Rigidity theory generically locally rigid in dimension 3 if and only if the multigraph 5G, obtained from G by multiplying each edge of G by 5, contains 6 edge-disjoint spanning trees.

In dimension one, it is a well known fact that a graph is generically globally rigid if and only if it is 2-connected. The problem of combinatorially characterizing 2-dimensional generically globally rigid graphs was settled by Jackson and Jord´an. A graph is redundantly generically locally rigid in dimension d if deleting any edge results in a generically locally rigid graph in dimension d.

Theorem 3.3.9 (Hendrickson [53], Jackson and Jord´an [58]). A graph is generically globally rigid in dimension 2 if and only if it is 3-connected and redundantly generically locally rigid in dimension 2.

In fact, the “only if” part of this theorem is shown by Hendrickson [53] for all dimensions. He conjectured the truth of the converse for all dimensions. Jackson and Jord´an conﬁrm it for dimension 2 by proving an inductive construction of graphs that are 3-connected and redundantly locally rigid in the plane: these graphs are constructed from K4 by a sequence of edge-additions and 2-dimensional 1-extensions, which are known to preserve the generic global rigidity. For d = 3,

Connelly proved that K5,5 is a counter example. Though inﬁnite families of counter examples for Hendrickson’s conjecture in higher dimensions (d ≥ 5) are obtained 3 [40], it remains an open question whether K5,5 is the only counter example in R . Unlike generic local rigidity and global rigidity, knowledge on combinatorial properties of generically universally rigid graphs is quite modest even in dimension one. Probably, the only known construction to create generically universally rigid graphs is the following.

Lemma 3.3.10 (Ratmanski [88]). A graph G on at least d + 2 vertices is d- dimensional generically universally rigid (d-GUR) if G can be obtained from Kd+1 by the following operations:

(i) add an edge,

(ii) choose two graph G1,G2 built by these operations, choose two sets U1,U2 of

each with |U1| = |U2| ≥ d + 1, delete all edges joining vertices of U1 in G1,

then glue the two graphs together along the vertices in U1 and U2.

In particular, if we add a vertex to a d-GUR graph G and connect it to at least d + 1 vertices of G then we obtain a d-GUR graph. It is an open question whether the converse of Lemma 3.3.10 is true.

49 3.3. Combinatorial rigidity

Figure 3.8: A 5-connected graph that is not generically rigid in R2 [80].

An interesting question in combinatorial rigidity theory is that whether high vertex-connectivity implies generic local/global rigidity. For dimension 2, based on Laman’s characterization, Lovász and Yemini [80] showed that every 6-vertex- connected graphs minus any three edges are generically locally rigid in R2. They also pointed out that 6 is the minimum value, i.e., there are 5-vertex-connected graphs that are not generically locally rigid in R2 (Figure 3.8). Combining the result of Lovász and Yemini with Theorem 3.3.9, it yields that 6-vertex-connected graphs are generically globally rigid in R2. In dimension d ≥ 3, as for characterization of generic local rigidity, the relation between vertex-connectivity and generic local/global rigidity of graphs is unknown. It is conjectured by Lovász and Yemini that d(d + 1)-connectivity is sufficient. For molecular graphs, using the known combinatorial characterization proved by Katoh and Tanigawa (Theorem 3.3.8), Jordán [69] derived that every 7-vertex-connected molecular graphs are generically locally rigid in R3. The same question about the relation between vertex-connectivity and the generical universal rigidity of graphs would be asked. However, in Chapter 8 we prove that no vertex-connectivity can guarantee the generic universal rigidity of graphs in any dimension. In fact, we show that every complete bipartite graph, except K2, is non generically universally rigid in any dimension.

50 Chapter 4

Matroid approach

Contents 4.1 Introduction ...... 52 4.2 Characterizing abstract rigidity matroids ...... 53 4.3 1-extendable abstract rigidity matroids ...... 54 4.4 Submodular functions inducing ARMs ...... 58 4.5 A potential application ...... 61

51 4.1. Introduction

4.1 Introduction

The concept of abstract rigidity matroid was first introduced by Graver [46] as a generalization of the generic rigidity matroid. Yet application of abstract rigidity matroids in the study of rigidity is still limited, viewing the generic rigidity matroid as an abstract rigidity matroid allows one to concentrate on its combinatorial nature. On the other hand, abstract rigidity matroids are an interesting topic in its own right with many open questions and may have applications in other problems as discussed in Section 4.5. Before going into the definition of abstract rigidity matroids let us keep in mind the two simple but important properties of d-dimensional generic bar-joint frameworks whose detailed proof can be found in [47]. The first one is that, if two frameworks are glued together over at most d − 1 joints, then the composed framework is not rigid (it always allows a relative rotation between the two parts about a d − 2-dimensional affine space containing the glued joints). Moreover, if we add any bar crossing these two parts then the framework becomes “more rigid”. The second one is that, if two rigid frameworks are glued together over at least d joints, then the obtained framework is also rigid. We can formalize these observations as follows. In this chapter we abuse notation K(V ) to denote also the edge set of the complete graph on a vertex set V and Kt to denote the edge set of a complete graph on t vertices. Let Gd(n) be the generic rigidity matroid on the complete graph K = (V,K(V )) and r its rank function. An edge set E ⊆ K(V ) is said to be rigid if cl(E) = K(V (E)), i.e., E spans K(V (E)). In matroid language, the two properties above are restated as follows.

(C1) If |V (E) ∩ V (F )| ≤ d − 1, then cl(E ∪ F ) ⊆ K(V (E)) ∪ K(V (F )).

(C2) For every pair of rigid subsets E,F of K(V ), if |V (E) ∩ V (F )| ≥ d, then E ∪ F is rigid.

Let Ad be a matroid on K(V ) with closure operator clAd (·). A subset E of

K(V ) is said to be rigid (in Ad) if clAd (E) = K(V (E)). Then Ad is called a d-dimensional abstract rigidity matroid if it satisﬁes (C1) and (C2). This chapter presents an approach to the problem of characterizing the generic rigidity matroid from a pure matroid viewpoint. First, in Section 4.2, we describe our result on combinatorial characterization of abstract rigidity matroids. Then, in Section 4.3, we introduce the concept of 1-extendable abstract rigidity matroid – a generalization that captures more properties of the generic rigidity matroid than

52 Chapter 4. Matroid approach abstract rigidity matroids – and show that although in dimension 2 a 1-extendable abstract rigidity matroid coincides with the generic rigidity matroid, in dimension 3 they can be diﬀerent. Section 4.4 is devoted to the study of intersecting submodular functions that induce abstract rigidity matroids. We provide a necessary condition for these functions. With an additional assumption on the symmetry, we show that this necessary condition is also suﬃcient. We close the chapter with a discussion on a potential application of our results on abstract rigidity matroids. This work originates from the author’s master’s thesis and partly published in [85] before the enrolment to the PhD program. We include these parts in this thesis to provide a complete view of the early approach. So many details and proofs in these parts will be omitted.

4.2 Characterizing abstract rigidity matroids

A vertex star of the complete graph (V,K(V )) is the set of all edges incident to some vertex v ∈ V . In [47], Graver, Servatius and Servatius posed two questions on the characterization of abstract rigidity matroids in dimension 2.

Question 1 [47, page 107] Is it true that a matroid M on the edge set of the complete graph (V,K(V )) is a 2-dimensional abstract rigidity

matroid if and only if all of the K4’s are circuits and all of the vertex stars minus an edge are cocircuits?

Question 2 [47, page 108] Is it true that a matroid M on the edge set of the complete graph (V,K(V )) is a 2-dimensional abstract rigidity

matroid if and only if all of the K4’s are circuits and r(K(U)) = 2|U|−3 for all U ⊆ V with |U| ≥ 2?

Subsequently, in [48], they gave an affirmative answer to Question 1 together with its generalization in higher dimension. In [85], we show that the condition in Question 1 and the one in Question 2 are both equivalent to the property that M is an abstract rigidity matroid. This gives an affirmative answer to Question 2 and its generalization as well as an alternative proof to the result of Graver, Servatius and Servatius [48, Theorem 0.2]. As a byproduct, we obtain a polynomial algorithm for testing if a matroid given by an independence oracle is a d-dimensional abstract rigidity matroid for any fixed d.

For E ⊆ K, v ∈ V \V (E) and u1, . . . , uk ∈ V (E), we call F = E+vu1 +···+vuk a k-valent 0-extension of E. Let Kt denote the edge set of a complete subgraph

53 4.3. 1-extendable abstract rigidity matroids on t vertices of (V,K). The principal ingredient to prove our characterization of abstract rigidity matroids is the following lemma.

Lemma 4.2.1 ([85]). A matroid on the edge set of the complete graph (V,K) is a d-dimensional abstract rigidity matroid if and only if it satisﬁes:

1. rAd (K(V )) = d|V | − d(d + 1)/2;

2. Each k-valent 0-extension of an independent set of Ad is also an independent

set of Ad for every k ≤ d.

The following theorem answers the two questions above and provides characterizations of d-dimensional abstract rigidity matroids for any d ≥ 2.

Theorem 4.2.2 ([85]). The following statements are equivalent for a matroid Ad on K(V ).

(i) Ad is a d-dimensional abstract rigidity matroid on K.

(ii) All Kd+2’s in K(V ) are circuits of Ad and all vertex stars minus (d−1) edges

are cocircuits of Ad.

(iii) All Kd+2’s in K(V ) are circuits of Ad and rAd (K(U)) = d|U| − d(d + 1)/2 for every U ⊆ V with |U| ≥ d + 1.

(iv) All Kd+2’s in K(V ) are circuits of Ad and rAd (K) = d|V | − d(d + 1)/2.

Theorem 4.2.2 implies that we can discern whether a matroid Ad given by an independence oracle is a d-dimensional abstract rigidity matroid for a ﬁxed positive integer d in polynomial time by checking condition (iv). Although it is not mentioned in [48], condition (ii) also implies a polynomial time algorithm for testing whether a given matroid is a d-dimensional abstract rigidity matroid. However, to check condition (ii), we would need to verify whether every vertex star minus (d − 1) edges is a cocircuit, or, equivalently, its complement is a hyperplane, which would take O(nd+1) calls to the independence oracle, while an algorithm using condition (iv) would need only O(nd) oracle calls.

4.3 1-extendable abstract rigidity matroids

In this section, extensions on a graph are regarded as extensions on its edge set. A matroid M on the edge set K of the complete graph (V,K) is called a d-dimensional 1-extendable abstract rigidity matroid if M is a d-dimensional abstract rigidity

54 Chapter 4. Matroid approach

Figure 4.1: The graph H in the proof of Theorem 4.3.1. Each vertex is connected to four

nearest vertices and its opposite vertex. |H| = 30 = r(K12) in G3(12).

v k+1 vm vk+1

v m v vertex v v v k v’ k splitting

v v v1 v2 1 2

Figure 4.2: Vertex splitting operation matroid on K and if F ⊆ K is a d-dimensional 1-extension of an independent set in M then F is independent in M. The generic rigidity matroid on K is an example of a 1-extendable abstract rigidity matroid on K. In dimension 2, the generic rigidity matroid is characterized by Laman’s condition and also by Theorem 3.3.6. A corollary of these characterizations is that the 2-dimensional generic rigidity matroid is the unique 2-dimensional 1-extendable abstract rigidity matroid (Graver, Servatius and Ser- vatius [47, Theorem 4.2.3]). The following theorem shows that G3 is not the only 3-dimensional 1-extendable abstract rigidity matroid.

Theorem 4.3.1. There exists a 3-dimensional 1-extendable abstract rigidity matroid that is not a generic rigidity matroid.

We brieﬂy describe the idea of the proof of Theorem 4.3.1. Let us consider the subset H of K = K12 depicted in Figure 4.1. ′ For a subset E ⊂ K with vv1, vv2, . . . , vvm ∈ E and a vertex v ∈ V \ V (E) ′ ′ ′ ′ ′ the edge set F = E + vv + v v1 + v v2 − vvk+1 − · · · − vvm + v vk+1 + ··· v vm with 2 ≤ k ≤ m is said to be obtained from E by a vertex splitting operation (Figure

4.2). In G3, if E is independent, then F is also independent (Whiteley [108]).

The graph H can be obtained from K4 by a vertex splitting operation and a

55 4.3. 1-extendable abstract rigidity matroids

vertex 1−ext. splitting

1−ext.

1−ext. 1−ext.

0−ext.

0−ext. 0−ext.

Figure 4.3: Building H from K4 by a sequence of 0-extensions, 1-extensions and a vertex splitting operation. New vertices are denoted in black.

sequence of 1-extensions and 0-extensions as shown in Figure 4.3. Since K4 is independent in the 3-dimensional generic rigidity matroid, E(H) is also independent in G3(12).

Furthermore, we can show that the subset H − e + f is a base of K in G3(12) for every e ∈ H and f ∈ K \ H. Figure 4.4 illustrates how a graph H + e − f can be constructed from K4 by 0-extensions and 1-extensions. ′ Now, let B be the set of bases of G3(12), then H ∈ B. Let B = B − H. Then using the fact that E(H) − e + f is a base of K in G3(12) for every e ∈ H and f ∈ K \ E(H) we can show that B′ is a set of bases of a matroid M′ on K. This matroid M′ is obviously a 1-extendable abstract rigidity matroid since the only ′ base that we delete from G3(12) to obtain M is the edge set of a graph with all vertices of degree 5. Remark: Walter Whiteley communicated that, in dimension d ≥ 4, it had been already known that the spline matroid is a 1-extendable abstract rigidity matroid which is distinct from the generic rigidity matroid. In dimension 3, however, the spline matroid is conjectured to be isomorphic to the rigidity matroids [110]. Moreover, the example in the proof of Theorem 4.3.1 is also showing that the inde-

56 Chapter 4. Matroid approach

1−ext. 1−ext.

1−ext.

1−ext. 1−ext.

0−ext.

1−ext. 0−ext.

Figure 4.4: Building an edge set H − e + f from K4 by a sequence of 0-extensions and 1-extensions. New vertices are denoted by black nodes.

57 4.4. Submodular functions inducing ARMs pendence preservation of vertex splitting operation is not a matroidal consequence of 1-extendability. Using the graph K6,6 minus a perfect matching one can show that the independence preservation of X-replacement operation is not a matroidal consequence 1-extendability and the independence preservation of vertex splitting operation.

4.4 Intersecting submodular functions inducing abstract rigidity matroids.

Let us recall that an integer-valued function f : 2S → Z is called an intersecting submodular function if f(X) + f(Y ) ≥ f(X ∪ Y ) + f(X ∩ Y ) for every X,Y ⊆ S with X ∩ Y =6 ∅. In this section when we talk about intersecting submodular functions, we mean non-decreasing intersecting submodular functions. Recall also that an intersecting submodular function f : 2S → Z induces a matroid with the collection of independent sets

I(f) = {I ⊆ S | f(J) ≥ |J|, ∀J ⊆ I,J =6 ∅}.

We say that a matroid M on S is induced by intersecting submodular function f if the collection of independent sets of M coincides with I(f). Note that diﬀerent intersecting submodular functions can induce the same matroid. Laman’s condition can be restated as follows.

The intersecting submodular function f deﬁned by f(X) = 2|V (X)|−3,

for ∅ 6= X ⊆ K, induces G2(n).

A natural question is: What are necessary and suﬃcient conditions for an intersecting submodular function to induce Gd(n)? We can also consider a relaxed version on the conditions for an intersecting submodular function to induce an abstract rigidity matroid. In the following, we will show that all intersecting submodular functions inducing an abstract rigidity matroid must have the same value as the rank function of the rigidity matroid on the edge set of a complete subgraph of K(V ). Conversely, this condition together with the “symmetry” will ensure that the induced matroid is an abstract rigidity matroid.

Theorem 4.4.1. Let f : 2K → Z be an intersecting submodular function that induces a d-dimensional abstract rigidity matroid M on K. Let Kt ⊆ K be the

58 Chapter 4. Matroid approach

edge set of a complete subgraph on t vertices for t ≥ 1. Then f(Kt) = dt−d(d+1)/2 holds for t ≥ d + 2.

We say that a function f : 2K → Z is iso-symmetric if f(E) = f(F ) whenever E,F ⊆ K induce two isomorphic subgraphs.

K Theorem 4.4.2. Suppose that f : 2 → Z satisﬁes f(Kt) = dt − d(d + 1)/2 for t ≥ d + 2, f(Kt) ≥ |Kt| for t ≤ d + 1, for the edge set Kt of any complete subgraph on t vertices of K. Suppose further that f is iso-symmetric. Then f induces a d-dimensional abstract rigidity matroid on K.

For the sake of simplicity we will demonstrate the two theorems above for the 2-dimensional case. The same arguments work for higher dimensions.

Proof of Theorem 4.4.1.

We prove by induction on t. Let K4 ⊆ K be the edge set of a complete subgraph on 4 vertices. Since K4 is a circuit of the abstract rigidity matroid M (by Theorem

4.2.2) and f induces M we have f(K4) < |K4| and f(K4 − e) ≥ |K4 − e| for all e ∈ K4. Therefore, f(K4) = |K4|−1 = 2×4−3 holds. Suppose that f(Kt) = 2t−3 holds for some t ≥ 4.

Claim 4.4.3. Let v1, v2 ∈ V (Kt), v ∈ V (K)\V (Kt). Then f(Kt + vv1 + vv2) = f(Kt) + 2. Proof. We have

f(Kt + vv1 + vv2) ≥ rM(Kt + vv1 + vv2)

= rM (Kt) + 2 (by Lemma 4.2.1)

= f(Kt) + 2 (by induction hypothesis).

Suppose on contrary that f(Kt + vv1 + vv2) ≥ f(Kt) + 3. Let v3 be a vertex in V (Kt) − {v1, v2} and Bt a base of Kt. We prove that Bt + vv1 + vv2 + vv3 is independent in M, which has greater rank than Kt+1, a contradiction. We just need to prove that f(X) ≥ |X| for every subset X of Bt + vv1 + vv2 + vv3. If

{vv1, vv2, vv3} * X or X = {vv1, vv2, vv3} then evidently X is independent in M by Lemma 4.2.1, thus f(X) ≥ |X| holds. Now, if X = Xt + vv1 + vv2 + vv3 with

∅ 6= Xt ⊆ Bt, then

f(X) + f(Bt) ≥ f(X ∪ Bt) + f(X ∩ Bt)

= f(Bt + vv1 + vv2 + vv3) + f(Xt)

≥ f(Bt) + 3 + |Xt|.

59 4.4. Submodular functions inducing ARMs

It implies that f(X) ≥ |X| holds.

The following claim completes the proof of Theorem 4.4.1 for d = 2.

Claim 4.4.4. Let k be an integer such that 2 ≤ k ≤ t. Suppose that v1, . . . , vk ∈

V (Kt) and v∈ / V (Kt). Then f(Kt + vv1 + ··· + vvk) = f(Kt) + 2 holds.

Proof. We prove by induction on k. When k = 2 the statement holds by Claim

4.4.3. For k = 3, if f(Kt + vv1 + vv2 + vv3) ≥ f(Kt) + 3 then using the same argument as that in Claim 4.4.3 we deduce a contradiction. Thus the statement holds for k = 3. Now suppose that the statement holds for some k ≥ 3. Let vk+1 be a vertex in V (Kt) − {v1, . . . vk}, then using the intersecting submodularity of f we have

f(Kt + vv1 + vv2 + ··· + vvk + vvk+1) + f(Kt + vv2 + ··· + vvk)

≤ f(Kt + vv1 + ··· + vvk) + f(Kt + vv2 + ··· + vvk+1), which implies that f(Kt + vv1 + vv2 + ··· + vvk + vvk+1) ≤ f(Kt) + 2 and thus f(Kt + vv1 + vv2 + ··· + vvk + vvk+1) = f(Kt) + 2.

Proof of Theorem 4.4.2.

Let M(f) be the matroid induced by f. Let Kt be the edge set of an arbitrary complete subgraph on t vertices of K.

Claim 4.4.5. If v∈ / V (Kt) and v1 ∈ V (Kt) then f(Kt + vv1) ≥ f(Kt) + 1 holds.

Proof. Assume on the contrary that f(Kt + vv1) = f(Kt). Then, by the iso- symmetry of f, f(Kt + vvi) = f(Kt) holds for every vi ∈ V (Kt). Using the intersecting submodularity of f we can easily derive that f(Kt) = f(Kt + vv1 +

··· + vvt), where v1, . . . , vt are all the vertices of Kt. Then the edge set Kt+1 =

Kt +vv1 +···+vvt of the complete subgraph on t+1 vertices satisﬁes 2(t+1)−3 = f(Kt+1) = f(Kt) = 2t − 3, which is a contradiction.

Claim 4.4.6. If v∈ / V (Kt) and v1, v2 ∈ V (Kt) then f(Kt +vv1 +vv2) ≥ f(Kt)+2.

Proof. From Claim 4.4.5 we have f(Kt + vv1 + vv2) ≥ f(Kt) + 1. Assume on the contrary that f(Kt +vv1 +vv2) = f(Kt)+1. Using induction on k, the intersecting submodularity and the iso-symmetry of f we can deduce that f(Kt + vv1 + ··· + vvk) = f(Kt) + 1 holds for every k and v1, . . . , vk ∈ V (Kt). In particular, the edge set Kt+1 = Kt + vv1 + ··· + vvt of the complete subgraph on t + 1 vertices satisﬁes

2(t + 1) − 3 = f(Kt+1) = f(Kt) + 1 = 2t − 2, a contradiction.

60 Chapter 4. Matroid approach

Claim 4.4.7. A 1-valent 0-extension of an independent set of M(f) is again an independent set of M(f).

Proof. Let Bt be a base of Kt in matroid M(f), v∈ / V (Kt) and v1 ∈ V (Kt). Then f(Bt + vv1) = f(Kt + vv1) ≥ f(Kt) + 1 = f(Bt) + 1, by Claim 4.4.5. Let X be an arbitrary non-emty subset of Bt. By the intersecting submodularity

f(X + vv1) + f(Bt) ≥ f((X + vv1) ∪ Bt) + f((X + vv1) ∩ Bt)

= f(Bt + vv1) + f(X)

≥ f(Bt) + 1 + |X|.

Thus, f(X + vv1) ≥ |X + vv1| holds. It follows that Bt + vv1 is independent in the matroid M(f), which implies the statement of Claim 4.4.7

Claim 4.4.8. A 2-valent 0-extension of an independent set in M(f) is again an independent set in M(f).

Proof. Let Bt be a base of Kt in the matroid M(f), v∈ / V (Kt) and v1, v2 ∈ V (Kt).

Assume on the contrary that there exists a subset X of Bt + vv1 + vv2 such that

X is a circuit in M(f). Then, by Claim 4.4.7, X = Xt + vv1 + vv2 with Xt ⊆ Bt. Then, by the intersecting submodularity of f,

f(X) + f(Bt) ≥ f(X ∪ Bt) + f(X ∩ Bt)

= f(Bt + vv1 + vv2) + f(Xt)

= f(Kt + vv1 + vv2) + f(Xt)

≥ f(Bt) + 2 + |Xt| (by Claim 4.4.6).

Therefore, f(X) ≥ |X| holds, a contradiction.

Claim 4.4.7, Claim 4.4.8 and Theorem 4.2.2 imply that M(f) is a 2-dimensional abstract rigidity matroid on K, which completes the proof of Theorem 3.7 for d = 2.

4.5 A potential application

We end this chapter by discussing a potential application of our results on abstract rigidity matroids. Thomassen [103] conjectured that there exists a function f(k) such that every f(k)-connected graph has a k-connected orientation. Jord´an [68] conﬁrmed this

61 4.5. A potential application conjecture for k = 2 by showing that f(2) ≤ 18. This upper bound is improved by Cheriyan, Durand de Gevigney and Szigeti [18] to f(2) ≤ 14, using a similar idea. For k ≥ 3 no upper bound has been obtained. The main idea of Jordán [68] is that a 18-connected graph contains 3 edge-disjoint 2-connect spanning subgraphs. Using these 2-connected subgraphs he deduces a 2-connected orientation of the original graph. Thus comes the question if we can pack many k-connected spanning subgraphs in a highly connected graph. The 2-connected graphs used by Jordán are in fact 2-dimensional generically rigid graphs and the packing is obtained by using the rank formula given by the intersecting submodular function f(F ) = 2|V (F )| − 3 which induces G2. Therefore, we are interested in the question of finding a matroid such that the “rigid” graphs with respect to this matroid are k-connected, and that it possesses a “simple” inducing intersecting submodular function. k-dimensional abstract rigidity matroids may be good candidates since rigid graphs with respect to these matroids are k-connected and the sufficient condition for an intersecting submodular function to induce an abstract rigidity matroid is simple as shown in Section 4.4.

62 Chapter 5

Inductive constructions and decompositions of graphs

Contents 5.1 Introduction ...... 65 5.2 Graded sparse graphs ...... 68 5.2.1 Introduction ...... 68 5.2.2 A reduction theorem for graded sparse graphs ...... 70 5.2.3 Inductive construction of graded sparse graphs . . . . . 75 5.2.4 Decomposition of graded sparse graphs ...... 75 5.2.5 Graded sparse matroids ...... 81 5.3 (b, l)-sparse graphs ...... 86 5.3.1 Inductive construction ...... 86 5.3.2 (b, l)-pebble games ...... 93 5.4 Packing of matroid-based arborescences ...... 96 5.4.1 Introduction ...... 96 5.4.2 Proof of the main theorem ...... 102 5.4.3 Polyhedral description ...... 105 5.4.4 Algorithmic aspects ...... 106

63 5.4.5 Further remarks ...... 107

64 Chapter 5. Inductive constructions and decompositions

5.1 Introduction

This chapter presents combinatorial optimization results developed to solve the problem of characterizing the underlying graphs of infinitesimally rigid generic frameworks of many types. The starting point is the following fundamental result of Nash-Williams. Theorem 5.1.1 (Nash-Williams [83]). A graph G = (V,E) can be decomposed into m edge-disjoint spanning trees if and only if i(X) ≤ m|X| − m for every non empty subset X of V and |E| = m|V | − m. Recall that a graph G verifying the condition in the theorem above is called an (m, m)-tight graph. An inductive construction for (m, m)-tight graphs is also implicit in [83]. In fact, a typical proof that a graph has an infinitesimally rigid realization if and only if it has a spanning tight sparse subgraph, proceeds along the following lines: (a) Construct a rigidity matrix whose rank determines the infinitesimal rigidity of a framework; (b) Deduce that the existence of a tight sparse subgraph is a necessary condition for infinitesimal rigidity; (c) Use either an inductive construction, or a decomposition, of a tight sparse subgraph to demonstrate sufficiency by constructing a realization whose rigidity matrix attains the maximum rank. An inductive construction for (2, 3)-sparse graphs suggested by Henneberg is used to prove Laman’s fundamental result (see [10, 47, 58]. Similarly, Tay [99] used the inductive construction for (m, m)-tight graphs due to Nash-Williams [82], to d+1 show that when m = 2 , these graphs are exactly the underlying graphs of minimally infinitesimally rigid d-dimensional generic body-bar frameworks. Inductive techniques were also employed successfully by Katoh and Tanigawa [72] to settle the long-standing Molecular Conjecture (on rigidity of panel-hinge frameworks). The alternative approach in (c) is to use a decomposition of tight sparse graphs to give a direct construction of an infinitesimally rigid realization. Examples of this approach are Tay [98], Whiteley [107], and Jackson and Jordán [60]. The above mentioned result of Nash-Williams [82] was not motivated by rigidity theory. However, the applications mentioned above have stimulated interest in generalizing the results of Nash-Williams and Tutte.

65 5.1. Introduction

The first way to generalize these results is to consider (m, l)-tight graphs. Fekete and Szeg˝o[33] provide an inductive construction of (m, l)-tight graphs for 0 ≤ l ≤ m. Whiteley [109] deduces a decomposition of (m, l)-tigh graph for 0 ≤ l ≤ m from matroid decomposition. Haas [51] characterizes (m, l)-tight sparse graph for 0 ≤ m ≤ l < 2m − 1 in terms of an “lT m-tree decomposition”, a concept rooted in Crapo’s work on decomposition of Laman’s graphs [26]. These decompositions are re-obtained by Streinu and Theran [96] using pebble games. The study of frameworks with different kinds of constraints led Lee, Streinu and Theran [79] to consider a more generalized class of graphs in which different types of edges satisfy different sparsity conditions. They call these graphs graded sparse graphs. As an example they consider the bar-joint-slider model in which bars are represented by non-loop edges and sliders are represented by loops. They show that a graph can be realized as a rigid slider-bar-joint 2-dimensional framework if and only if it has a subgraph H which is (2, 0)-tight sparse and the subgraph of H induced by the non-loop edges is (2, 3)-sparse. In section 5.2, we provide an inductive construction as well as a decomposition for graded tight sparse graphs. The decomposition will be used in Chapter 6 to characterize different types of body- length-direction frameworks as well as to re-obtain a result of Katoh and Tanigawa on the characterization of underlying graphs of generic body-bar frameworks with bar-boundary. We also propose a second way to generalize the result of Nash-Williams by considering (b, l)-tight sparse graphs. These tight sparse graphs arise in the rigidity context when one considers frameworks that contain bodies of different dimensions. An inductive construction for a special instant of b, l was derived by Tay [98] to characterize body-rod-bar frameworks. In section 5.3, we give an inductive construction of (b, l)-tight graphs for l not greater than the minimum value of b. We also characterize (b, l)-sparse graphs as resulting graphs in (b, l)-pebble games, a generalization of (k, l)-pebble games by Lee, Streinu [78]. A third way of generalization is proposed by Katoh and Tanigawa in [73]. They regard each tree as a tree with a root r and replace the condition that each vertex is covered by all the m trees 1 with a matroidal condition on the root set of trees covering each vertex. Motivated by the study of bar-joint-slider frameworks, their result is applied successfully to characterize several other types of frameworks with boundaries (cf. Section 6.2). In Section 5.4, we derive a directed counterpart to

1This condition is equivalent to the condition that the m trees in the decomposition/packing are spanning.

66 Chapter 5. Inductive constructions and decompositions the result of Katoh and Tanigawa. Furthermore, we show that our directed result implies their undirected result. As a consequence, we obtain a shorter proof for Katoh and Tanigawa’s result.

67 5.2. Graded sparse graphs

5.2 Graded sparse graphs

5.2.1 Introduction

Let G = (V,E) be a graph. An r-grading of E is a strictly decreasing sequence of sets (E1,...,Er) with E = E1 ⊃ E2 · · · ⊃ Er. A graph G with an r-grading

(E1,...,Er) is called an r-graded graph, or simply a graded graph when the value of r is clear. The grade of an edge e ∈ E, denoted by σe, is the maximum number t such that e ∈ Et. Let m be a positive integer and d = (d1, . . . , dr) be an r-tuple of integers with 0 ≤ d1 ≤ · · · ≤ dr ≤ 2m − 1. The r-graded graph G is said to be (m, d)-graded sparse if the subgraph Gt = (V,Et) is (m, dt)-sparse for every t = 1, . . . , r, and (m, d)-graded tight if in addition it has |E| = m|V | − d1. For a subset X of V , we denote by it(X) the number of edges of Et induced by X. The graded sparsity condition can then be written as

it(X) ≤ m|X| − dt, for all non-empty X ⊆ V and all t = 1, . . . , r.

We will need some more definitions and notations to state our main results on graded sparse graphs. A splitting off at a vertex v of G is the operation which deletes two edges f = vx and g = vy incident to v and adds a new edge e = xy of grade σe = min{σf , σg}. When f is a loop at v the splitting off just deletes f and changes the grade of g to min{σf , σg}. We will refer to such a splitting off as a loop replacement. A splitting off for which neither f nor g is a loop will be referred to as a proper splitting off. A(k, ℓ)-reduction of an (m, d)-graded tight graph G is defined as follows. We first choose a vertex v which is incident with m+k edges in total, including exactly ℓ loops, for some 0 ≤ k ≤ m − ℓ. If k = 0 then we simply delete v from G. If k ≥ 1 then we perform a sequence of ℓ loop replacements followed by k proper splitting offs at v, then delete v and the remaining m − k − ℓ edges incident to it. A (k, ℓ)-reduction at v is admissible if the resulting graph is also (m, d)-graded tight. The inverse operation to a (k, ℓ)-reduction is called a (k, ℓ)-extension. A (k, ℓ)- extension of H is an r-graded graph obtained from H by the following steps.

(a) Delete k edges ei = xiyi, 1 ≤ i ≤ k, from H.

(b) Add a new vertex v and m − k − ℓ new edges of arbitrary grades from v to the vertices of H.

68 Chapter 5. Inductive constructions and decompositions

(i) the minimum grade of fi, gi, hi,1, . . . , hi,ℓi is equal to σei ;

(ii) ℓ1 + ℓ2 + ... + ℓk = ℓ; (iii) the total number of loops of grade at least t added at v is less than or

equal to m − dt for all 1 ≤ t ≤ r.

In the special case when k = 0 we obtain a (0, ℓ)-extension by simply adding a new vertex v, ℓ loops incident with v and m − ℓ edges from v to H in such a way that condition (c)(iii) above is satisﬁed.

Contribution: Our ﬁrst result is that admissible reductions always exist, when

0 ≤ d1 ≤ · · · ≤ dr ≤ m.

Theorem 5.2.1. Suppose that m is a positive integer and d = (d1, . . . , dr) is a non-decreasing sequence of integers 0 ≤ d1 ≤ · · · ≤ dr ≤ m. Let G be an (m, d)- graded tight graph on at least two vertices. Then G has a vertex v which is incident with m + k edges, including exactly ℓ loops, for some 0 ≤ k ≤ m − ℓ. Furthermore, for any such vertex v, G has an admissible (k, ℓ)-reduction at v.

We will show that a (k, ℓ)-extension preserves the property of being (m, d)- graded tight. Combined with Theorem 5.2.1, this will imply the following inductive construction for (m, d)-graded tight graphs.

Theorem 5.2.2. Suppose that m is a positive integer and d = (d1, . . . , dr) a non- dereasing sequence of integers 0 ≤ d1 ≤ · · · ≤ dr ≤ m. Let G be an r-graded graph. Then G is (m, d)-graded tight if and only if G can be obtained from an (m, d)- graded tight graph on one vertex by a sequence of (k, ℓ)-extensions. Moreover, if d1 > 0 then we need only (k, ℓ)-extensions with k + ℓ < m.

A pseudoforest is a graph in which each connected component contains at most one cycle. A pseudoforest is tight if each of its components contains exactly one cycle. (Equivalently G is a pseudoforest if it is (1, 0)-sparse and is a tight pseudoforest if it is (1, 0)-tight.) An (m, d)-graded pseudoforest decomposition of an r-graded graph G is a partition of E(G) into m edge-disjoint pseudoforests F1,F2,...,Fm such that for each t, 1 ≤ t ≤ r, and all i, 1 ≤ i ≤ dt, the restriction of Fi on Et is a forest. Our third main result characterizes (m, d)-graded tightness in terms of (m, d)-graded pseudoforest decompositions.

69 5.2. Graded sparse graphs

Theorem 5.2.3. Let 0 ≤ d1 ≤ · · · ≤ dr ≤ m be integers and G an r-graded graph. Then G is (m, d)-graded tight if and only if G has an (m, d)-graded pseudoforest decomposition consisting of d1 spanning trees and m − d1 spanning tight pseudoforests.

Theorems 5.2.1, 5.2.2 and 5.2.3 will be proved in Sections 5.2.2, 5.2.3 and 5.2.4, respectively. Furthermore, in Section 5.2.5 we study the matroids whose independent sets are the edge sets of (m, d)-graded sparse graphs. We give a short direct proof for their matroid decomposition and thus obtain an alternative proof for the pseudoforest decomposition in Theorem 5.2.3.

Results in this section are from a joint work with Bill Jackson [66].

5.2.2 A reduction theorem for graded sparse graphs

We suppose throughout this section that

m is a positive integer and d = (d1, . . . , dr) with 0 ≤ d1 ≤ · · · ≤ dr ≤ m. (5.2.1)

Our aim is to prove Theorem 5.2.1. We ﬁrst show that every (m, d)-graded tight graph G has a vertex which is incident with the required number of edges.

Lemma 5.2.4. Let G = (V,E) be an (m, d)-graded tight graph. Then G has a vertex v which is incident with m + k edges for some 0 ≤ k ≤ m − i(v).

Proof. Let v be a vertex of minimum degree in G. Since G is tight, we have

d(x) = 2|E| = 2m|V | − 2d1 ≤ 2m|V |. x∈V X Hence d(v) ≤ 2m and so d(v) − i(v) ≤ m + m − i(v). On the other hand

m|V | − d1 = |E| = i(V − v) + d(v) − i(v) ≤ m(|V | − 1) − d1 + d(v) − i(v) so d(v) − i(v) ≥ m.

We next show that if v is a vertex of G with d(v) − i(v) = m + k for some 1 ≤ k ≤ m − i(v), then we can construct an (m, d)-graded sparse graph from G by performing a sequence of splitting offs at v (consisting of i(v) loop replacements followed by k proper splitting offs). We construct these splitting offs one at a time

70 Chapter 5. Inductive constructions and decompositions in such a way that we preserve (m, d)-graded sparseness as well as an additional sparsity condition at v. We assume for the remainder of this section that

H = (V,E) is an (m, d)-graded sparse graph with grading (E1,...,Er) (5.2.2) and v is a vertex of H with d(v) = m + k + c − 2s, i(v) = max{0, c − s}, c ≥ 0, k ≥ 1 and 0 ≤ s < k + c ≤ m. (5.2.3) We imagine that c is the number of loops incident to v in G and that H has been obtained from G by performing s splitting oﬀs at v. Noting that the number of edges incident to v is d(v) − i(v), condition (5.2.3) implies that

The number of edges incident to v is strictly greater than m − s. (5.2.4)

In fact, if i(v) = 0 then the number of edges incident to v is d(v) = m+k +c−2s = m − s + (k + c − s) > m − s by (5.2.3). If i(v) > 0 then, by (5.2.3), i(v) = c − s, so the number of edges incident to v is d(v) − i(v) = m + k + c − 2s − (c − s) = m + k − s > m − s since k ≥ 1. For X,Y ⊆ V , we denote the intersection of Et and the set of edges from X to Y by Et(X,Y ). (This set contains in particular all edges of Et induced by X ∩ Y .) We also use E(X,Y ) for E1(X,Y ), it(X,Y ) for

|Et(X,Y )| and i(X,Y ) for |E(X,Y )|.

We say that H is (v, s)-good if it(X) ≤ m|X| − dt − s for all X ⊆ V that properly contain v, and all 1 ≤ t ≤ r. We will suppose henceforth that

H is (v, s)-good. (5.2.5)

Our aim is to find a splitting off at v in H such that the new graph is both (m, d)-graded sparse and (v, s+1)-good. We call such a splitting off feasible. To do this we need to consider the circumstances in which a splitting off is not feasible. A nonempty set X ⊆ V − v is said to be t-critical, or simply critical, if it satisfies it(X) = m|X| − dt for some 1 ≤ t ≤ r. A set X ⊆ V is t-crucial, or simply crucial, if it properly contains v and satisfies it(X) = m|X| − dt − s. The grade σ(X) of a crucial set X is the maximum value of t for which X is t-crucial. The following result characterizes when the splitting off operation is feasible.

Lemma 5.2.5. Let f = vx and g = vy be two edges incident to v with σf ≤ σg. Then splitting oﬀ f and g at v is feasible if and only if

(a) no t-critical set contains both x and y for all 1 ≤ t ≤ σf , and

71 5.2. Graded sparse graphs

(b) all crucial sets X have σ(X) ≤ σg and X ∩ {x, y}= 6 ∅. In addition, if σ(X) >

σf , then y ∈ X.

Proof. This follows from the deﬁnitions of a feasible splitting oﬀ, and critical and crucial sets.

It follows that we will need to analyze the structure of the families of critical and crucial sets in order to show that there exists a feasible splitting oﬀ. Our basic tool is the following result.

Lemma 5.2.6. Suppose that X,Y ⊆ V and that t, t′ are integers with 1 ≤ t ≤ t′ ≤ r. Then

it(X) + it′ (Y ) + it(X \ Y,Y \ X) ≤ it(X ∪ Y ) + it′ (X ∩ Y ).

Proof. This follows easily by counting the contribution of each edge of H to both sides of the inequality.

Lemma 5.2.7. Suppose X is a t-critical set, Y is a t′-critical set, t ≤ t′ and ′ X ∩ Y =6 ∅. Then X ∪ Y is t-critical, X ∩ Y is t -critical and it(X \ Y,Y \ X) = 0.

Proof. Since H is (m, d) graded sparse, we have

it(X ∪ Y ) + it′ (X ∩ Y ) ≤ m|X ∪ Y | − dt + m|X ∩ Y | − dt′

= m|X| − dt + m|Y | − dt′

= it(X) + it′ (Y )

≤ it(X ∪ Y ) + it′ (X ∩ Y ) − it(X \ Y,Y \ X).

Hence equality must occur everywhere and so it(X ∪ Y ) = m|X ∪ Y | − dt, it′ (X ∪

Y ) = m|X ∪ Y | − dt′ and it(X \ Y,Y \ X) = 0.

An immediate consequence of Lemma 5.2.7 is the following.

Corollary 5.2.8. Suppose x ∈ V −v is contained in at least one critical set. Then the union of all critical sets which contain x is a t-critical set where t is the smallest grade of all the critical sets which contain x.

The proof of the following lemma is similar to that of Lemma 5.2.7.

Lemma 5.2.9. Suppose that X is a t-crucial set, Y is a t′-crucial set, t ≤ t′ and |X ∩ Y | ≥ 2. Then X ∪ Y is t-crucial and X ∩ Y is t′-crucial.

72 Chapter 5. Inductive constructions and decompositions

Lemma 5.2.10. If X is a t-critical set then it(v, X + v) ≤ m − s.

Proof. Since H is (v, s)-good, we have m(|X| + 1) − dt − s ≥ it(X + v) = it(X) + it(v, X + v) = m|X| − dt + it(v, X + v).

Therefore, it(v, X − v) ≤ m − s.

Lemma 5.2.11. If X is a t-crucial set then it(v, X) ≥ m−s and it(v, X −v) ≥ 1.

Proof. We have it(X−v)+it(v, X) = it(X) = m|X|−dt−s = m|X−v|−dt+m−s ≥ it(X−v)+m−s since H is (m, d)-graded sparse. Hence it(v, X) ≥ m−s. Suppose it(v, X −v) = 0.

Then m − s ≤ it(v, X) = it(v) ≤ i(v) = max{0, c − s}. Since m − s > 0 by (5.2.3), we have m − s ≤ c − s and m ≤ c. This contradicts the facts that m ≥ k + c and k ≥ 1 by (5.2.3).

A crucial set X is a minimal crucial set if it is not properly contained in any other crucial set.

Lemma 5.2.12. Suppose X and Y are distinct minimal crucial sets of grades t ′ ′ and t , respectively, where t ≤ t . Then X∩Y = {v}, E(v, V ) = Et(v, X)∪Et′ (v, Y ) and E(v) = Et′ (v).

Proof. The fact that X ∩ Y = {v} follows from Lemma 5.2.9 and minimality. By Lemma 5.2.11 and (5.2.3), m+k+c−2s = d(v) ≥ it(v, X)+it′ (v, Y ) ≥ 2(m−s) = m+m−2s ≥ m+k+c−2s.

Hence, equality must hold everywhere. In particular d(v) = it(v, X) + it′ (v, Y ), which implies that E(v, V ) = Et(v, X) ∪ Et′ (v, Y ) and E(v) = Et′ (v).

Lemma 5.2.13. There are at most two distinct minimal crucial sets in H, and the grade of a minimal crucial set is the maximum grade among all the crucial sets that contain it.

Proof. Suppose that X1,X2 and X3 are three distinct minimal crucial sets. Then

Xi ∩ Xj = {v} and Xi ∪ Xj contains all neighbours of v for all 1 ≤ i < j ≤ 3 by Lemma 5.2.12. This implies that all edges incident to v are loops, and contradicts the fact that i(v, Xi − v) ≥ 1 by Lemma 5.2.11. The second part of the lemma follows immediately from Lemma 5.2.9.

73 5.2. Graded sparse graphs

Lemma 5.2.14. There exists a feasible splitting oﬀ at v, and this can be taken to be a loop replacement when i(v) ≥ 1.

Proof. We ﬁrst consider the case when i(v) ≥ 1. Let f be a loop at v. Suppose that H has two minimal crucial sets X and Y , of grades t and t′, respectively, ′ where t ≤ t . Then f ∈ Et′ (v) by Lemma 5.2.12 and all crucial sets have grade at ′ most t by Lemma 5.2.13. We may choose an edge g ∈ Et′ (v, Y − v) by Lemma 5.2.11. Lemma 5.2.5 now tells us that splitting oﬀ f and g at v will be feasible. We next suppose that H has a unique minimal crucial set X, of grade t say. Then all crucial sets have grade at most t by Lemma 5.2.13 and it(v, X − v) ≥ 1 by Lemma

5.2.11. Lemma 5.2.5 now tells us that splitting oﬀ f with any edge g ∈ Et(v, X −v) will be feasible. When H has no crucial sets, Lemma 5.2.5 implies that splitting oﬀ f with any other edge at v will be feasible. Hence we may suppose that

i(v) = 0.

Let us consider the case that H has two minimal crucial sets X and Y of grades ′ t and t , respectively.We may choose edges f = vx ∈ Et(v, X − v) and g = vy ∈

Et′ (v, Y − v) by Lemma 5.2.11. If there is no critical set containing both x and y, splitting oﬀ f, g at v is feasible by Lemma 5.2.5. Suppose that there is a critical set containing both x and y. Let U be the maximal one and let t′′ be the minimal integer such that U is t′′-critical. Note that by Corollary 5.2.8, every critical set which contains x is a subset of U and t′′ is the smallest grade of all the critical sets containing x. By Lemma 5.2.10, it′′ (v, U + v) ≤ m − s. On the other hand, the number of edges incident to v is strictly greater than m − s by (5.2.4). Therefore, ′′ we can ﬁnd an edge h ∈ E(v, V − v) with either σh < t or h ∈ E(v, V − U − v).

This edge must belong to Et(v, X − v) or Et′ (v, Y − v) by Lemma 5.2.12. Without ′′ loss of generality, suppose that h ∈ Et′ (v, Y − v). If σh < t then splitting f, h at v is feasible by Lemma 5.2.5. If h = vz ∈ E(v, V − U − v) ∩ Et′ (v, Y − v), we claim that there is no critical set containing both x and z. In fact, if there is a critical set U ′ containing both x and z, by Lemma 5.2.7, U ∪ U ′ is a critical set properly containing x and y, which contradicts the maximality of U. Therefore, splitting oﬀ f, h at v is also feasible in this case by Lemma 5.2.5. We now consider the case that there exists at most one minimal crucial set. If there exists a crucial set, let X be the unique minimal crucial set and t its grade. Otherwise let X = V and t = 1. Then by Lemma 5.2.13, there is no crucial set of grade strictly greater than t. Let f = vx be an edge in Et(v, X − v). If there is a critical set containing x, let U be the maximal one and suppose that

74 Chapter 5. Inductive constructions and decompositions t′ is the smallest integer such that U is t′-critical. Note that then there is no t′′ ′′ ′ critical set containing x with t < t . By Lemma 5.2.10, it′ (v, U + v) ≤ m − s. On the other hand, the number of edges incident to v is strictly greater than m − s ′ by (5.2.4). Therefore, we can ﬁnd an edge h ∈ E(v, V − v) with either σh < t or h ∈ E(v, V − U − v). In both cases, splitting oﬀ f, h at v is feasible by the maximality of U and by Lemma 5.2.5.

5.2.3 Inductive construction of graded sparse graphs

We assume that (5.2.1) continues to hold throughout this section. Suppose that H is an (m, d)-graded tight graph and k, ℓ are non-negative integers with k + ℓ ≤ m.

Lemma 5.2.15. Suppose H is an (m, d)-graded tight graph and G is a (k, ℓ)- extension of H. Then G is (m, d)-graded tight.

Proof. We have |E(G)| = |E(H)| + m = m|V (H)| − d1 + m = m|V (G)| − d1. Hence it will suﬃce to show that G is (m, d)-graded sparse. Choose X ⊆ V (G) G H and t ∈ {1, 2, . . . , r}. If v 6∈ X then it (X) ≤ it (X) ≤ m|X| − dt since H is G (m, d)-graded sparse. If X = {v} then it (X) ≤ m − dt by condition (c)(iii) above. Hence we may suppose that v is properly contained in X. We adopt the notation used in the earlier deﬁnition of a (k, ℓ)-extension. Let bt be the number of edges H of {e1, e2, . . . , ek} which are contained in Et (X − v). These edges are deleted in G H Step (a) and hence it (X − v) = it (X − v) − bt ≤ m|X − v| − dt − bt. We add at most m − k − ℓ edges of Et at v in Step (b) and at most 2k + ℓ − (k − bt) edges of

Et at v in Step (c). Hence

G it (X) ≤ m|X − v| − dt − bt + m − k − ℓ + 2k + ℓ − (k − bt) = m|X| − dt.

Since this holds for all t, G is (m, d)-graded sparse.

Proof of Theorem 5.2.2

Suﬃciency follows from Lemma 5.2.15. Necessity follows by induction on |V (G)| and Theorem 5.2.1. 2

5.2.4 Decomposition of graded sparse graphs

In this section we give a proof of the decomposition of (m, d)-graded sparse graphs in Theorem 5.2.3 using the inductive construction. In Section 5.2.5 we give another

75 5.2. Graded sparse graphs proof of this result from matroid decomposition (Theorem 5.2.20). Since this graph decomposition is in fact equivalent to the matroid decomposition, the proof in this section serves also as an alternative proof for the matroid decomposition in Theorem 5.2.17. We will use Theorem 5.2.2 to construct this graph decomposition recursively. Let e = xy be an edge of an r-graded graph H and v be a vertex (not necessary in H). We call a v-subdivision of e the operation that deletes e from H and adds to

H two edges f = vx, g = vy with grades σf , σg satisfying σe = min{σf , σg}. The edges f, g are called subdividing edges of e. We call a v-subdivision proper if both f and g are non loops, otherwise we call it improper. Then the (k, ℓ)-extension can be seen as a sequence of k proper v-divisions and ℓ improper v-divisions followed by the addition of m − k − ℓ non loop edges from v to H.

Proof of Theorem 5.2.3

We prove by induction on the number of vertices. The decomposition is trivial when |V | = 1. Suppose that G is obtained from H by a sequence of k proper v-subdivisions and ℓ improper v-subdivisions followed by the addition of m − k − ℓ non loop edges from v to H. Our construction of the decomposition of G starts with a pseudoforest decomposition F of H, a set D = {e1, . . . , eh} of edges to be subdivided and a set A of edges which can be added to G. In fact, A =

{f1, g1, . . . , fh, gh, a1, . . . , am−h} where fi, gi are subdividing edges of ei, for i = 1, . . . , h. The sets F, A, D will be updated at each v-subdivision and edge addition. At a division of an edge e to f and g, we replace the pseudoforest F ∈ F that contains e with a pseudoforest F ′ by deleting e from F and adding to F some edges from A. We then remove the used edges from A and remove e from D. In fact, if the pseudoforest F has not already covered v then two edges from A are added to F , otherwise, only one is added. At an edge addition of an edge a, we simply choose an appropriate pseudoforest F from F and add a to F . Throughout the remaining of this section we suppose that we have a collection of m graded pseudoforests F = {F1,...,Fm}, a set D of edges to be subdivided and a set A containing the subdividing edges of edges in D as well as other edges to be added. Our construction of the decomposition will decrease |D| + |A|.

We will need some more notation. Let Ft denote {Fdt+1,...,Fm} for 1 ≤ t ≤ r. t Set F0 = F and Fr+1 = ∅. For 1 ≤ t ≤ r, we will use A to denote the subset of edges in A of grade at least t and for a pseudoforest F , we also use F t to denote the subgraph of F induced by the edges of grade at least t. When t = r + 1, At

76 Chapter 5. Inductive constructions and decompositions

t and F are just empty sets. Let ti = min{t : 1 ≤ t ≤ r, i ≤ dt} and ti = r + 1 if i > dr. We also set dr+1 = m. Note that, ti is the smallest t such that Fi ∈/ Ft. We restate this in the following convenient form.

Fi ∈ Ft if and only if t < ti. (5.2.6)

The following properties will be maintained during the construction of the pseudoforest decomposition.

t (P1) For F ∈ F \ Ft, F contains no cycle, for 1 ≤ t ≤ r.

t (P2) |{F ∈ Ft : F covers v}| + |{loops in A }| ≤ m − dt, for 1 ≤ t ≤ r.

(P3) |{F ∈ F : F covers v}| + |A| − |D| = m.

Note that at the very beginning, when F is the pseudoforest decomposition of H, these conditions hold, and when the construction terminates, |A| = |D| = 0 hence we obtain the desired decomposition.

Suppose that at a step in the construction, a pseudoforest Fi ∈ F is replaced ′ ′ ′ by a pseudoforest Fi and we obtain a new family F = F − Fi + Fi . To preserve the condition (P1) for the new family F ′ we only need to make sure that

′ ti (P1’) (Fi ) contains no cycle.

If |D| + |A| = 0 then the collection F is the desired decomposition. So suppose that |D| + |A| > 0.

First consider the case when D = ∅. If there is a loop at v then let c to be one of highest grade. By condition (P2), and since |Fσc | = m − dσc , there is an ′ ′ ′ Fi ∈ Fσc that does not cover v. Let Fi = Fi + c, A = A − c and D = D. Since ′ Fi does not cover v, Fi is a pseudoforest. Moreover, the only new cycle created by ′ ′ ti the operation in F is the loop c. Note that σc < ti by (5.2.6), so (Fi ) contains ′ ′ no cycle. Hence (P1’) holds for Fi and therefore (P1) holds for F . We check ′ condition (P2) for F . For t ≤ σc < ti,

′ |{F ∈ Ft : F covers v}| = |{F ∈ Ft : F covers v}| + 1, while |{loops in (A′)t}| = |{loops in At}| − 1,

77 5.2. Graded sparse graphs

′ ′ t thus (P2) holds for F . As for t > σc, we have (A ) = ∅ by the maximality of σc, ′ and |Ft| = m − dt. Thus

′ ′ t |{F ∈ Ft : F covers v}| + |{loops in (A ) }| ≤ m − dt for t > σc, thus (P2) also holds for F ′ in this case. (P3) also holds for F ′,A′,D′ since

|{F ∈ F ′ : F covers v}| = |{F ∈ F : F covers v}|+1, |D′| = |D|, and |A′| = |A|−1.

If there is no loop at v, let a be any edge in A and Fi be a pseudoforest that ′ ′ does not cover a (Fi exists by condition (P3)). Let Fi = Fi + a, A = A − a and D′ = D. (P1) holds for F ′ since no new cycle is created. (P2) also holds since t |{loops in A }| = 0 and |{F ∈ Ft : F covers v}| ≤ |Ft| = m−dt for 1 ≤ t ≤ r. (P3) holds for F ′,A′,D′ since |{F ∈ F ′ : F covers v}| = |{F ∈ F : F covers v}| + 1, |A′| = |A| − 1 and |D′| = |D|. Now suppose that D is not empty. Take an edge e = xy in D. Suppose that e is subdivided into f = vx, g = vy in A. Suppose also that e is an edge of an Fi in F.

Case 1: v is not covered by Fi. ′ ′ ′ ′ ′ Let Fi = Fi − e + f + g, F = F − Fi + Fi , D = D − e and A = A − f − g. ′ti First, we check condition (P1’). Suppose by contradiction that (Fi) contains a cycle C then, since (P1) holds for Fi, the cycle C must contain both f and g and ti ti ≤ min{σf , σg} = σe. Hence C − f − g + e is a cycle in Fi , a contradiction. Therefore (P1) holds for F ′. Condition (P3) also holds for F ′,D′,A′ since |{F ∈ F ′ : F covers v}| = |{F ∈ F : F covers v}|+1, |A′| = |A|−2 and |D′| = |D|−1. If F ′,A′ satisfy condition (P2) then we are done. So suppose that F ′,A′ violate (P2). ′ Since (P2) holds for F,A, there exists t < ti such that |{F ∈ Ft : F covers v}| + ′ t ∗ |{loops in (A ) }| > m−dt. Let t be the maximum t < ti such that this inequality holds. Then the following equality holds.

t∗ |{F ∈ Ft∗ : F covers v}| + |{loops in A }| = m − dt∗ , (5.2.7) keeping in mind that the sets of loops in A and that in A′ are the same as f, g are not loops. t∗ We claim that there exists a loop in A of grade strictly less than ti. Suppose ∗ by contradiction that |{loops in At }| = |{loops in Ati }|. We have,

ti |{F ∈ Fti : F covers v}| + |{loops of A }| ≤ m − dti . (5.2.8)

78 Chapter 5. Inductive constructions and decompositions

∗ Since Fi ∈ Ft \Fti and Fi does not cover v, we have,

∗ ∗ ∗ dti − dt = |Ft \Fti | > |F ∈ Ft \Fti : F covers v|. (5.2.9)

∗ However, from equalities (5.2.7) and (5.2.8) and the assumption |{loops in At }| = |{loops in Ati }|, we obtain

∗ ∗ ∗ |F ∈ Ft \Fti : F covers v| = |{F ∈ Ft : F covers v}|−|{F ∈ Fti : F covers v}| ≥ dti −dt , a contradiction to (5.2.9). Therefore our claim is true. ∗ t ti Let c be a loop in A \ A of maximum grade. Note that σc < ti and hence ′ ′ ′ Fi ∈ Fσc by (5.2.6). If in Fi − f the connected component containing g has a cycle ′′ ′ ′′ ′ ′′ ′′ ′′ then let Fi = Fi −g+c. Otherwise let Fi = Fi −f +c. Set F = F −Fi +Fi , A = A−f −c and D′′ = D−e. We claim that F ′′,A′′ and D′′ verify the conditions (P1), ′′ ′′ (P2), (P3). In fact, since v is already covered by Fi, |A | = |A| − 1, |D | = |D| − 1, (P3) obviously holds. To check (P1), first consider the case where the connected ′ ′ component containing g in Fi − f has a cycle. Then since Fi is a pseudoforest, the ′ connected component containing v in Fi −g has no cycle. Therefore, the connected ′′ ′ ′′ component of Fi = Fi − g + c containing v has only one cycle c and so Fi is a ′′ ′′ pseudoforest. Recall that σc < ti, we have that Fi satisfies (P1’) and hence F satisfies (P1). Now consider the case where the connected component containing ′ ′′ ′ ′′ g of Fi − f has no cycle. Then Fi = Fi − f + c is a pseudoforest and Fi satisfies ′′ ′′ (P1’) as σc < ti. It remains to check the condition (P2) for F ,A . Again, suppose by contradiction that there exists t such that

′′ ′′ |{F ∈ Ft : F covers v}| + |{loops in A |t}| > m − dt.

′′ ′ ′′t Then since |{F ∈ Ft : F covers v}| = |{F ∈ Ft : F covers v}|, and |{loops in A }| ≤ |{loops in A′t}|, it implies

′ ′ |{F ∈ Ft : F covers v}| + |{loops in A |t}| > m − dt.

∗ ∗ ′′ t By the maximality of t , we must have t ≤ t < ti. However, then |{loops in (A ) }| = ∗ |{loops in At}| − 1 holds, since the loop c belongs to At ⊆ At. Therefore, |{F ∈ ′′ ′′ t t Ft : F covers v}|+|{loops in (A ) }| = |{F ∈ Ft : F covers v}|+|{loops in A }| ≤ ′′ ′′ m − dt, a contradiction. We conclude that (P2) also holds for F ,A .

Case 2: v is already covered by Fi.

In this case we will replace the edge e = xy in Fi by one of the two edges f = vx, g = vy. We remove e from D and the replacing edge from D. Condition (P2) and (P3) then trivially hold.

79 5.2. Graded sparse graphs

Case 2.1: The subdivision is proper. ′ We will choose f or g to replace e so that Fi is a pseudoforest and satisﬁes (P1’).

If in Fi − e, one of x and y belongs to a diﬀerent connected component than ′ that contains v, without loss of generality, let it be x. Then Fi = Fi − e + f is a pseudoforest and (P1’) holds since no new cycle is created. So suppose that in Fi − e, both x and y belong to the same connected component as v. Then the unique cycle in the connected component of Fi containing v must contain e.

ti Therefore both Fi −e+f and Fi −e+g are pseudoforests. If both (Fi −e+f) and ti (Fi − e + g) contain a cycle then x and y are on the same connected component

ti ti of (Fi − e) and σe ≥ ti. Therefore, there exists a cycle that contains e in Fi . ti This contradiction to the assumption (P1) implies that either (Fi − e + f) or

ti (Fi − e + g) does not contain a cycle. Without loss of generality, suppose that ′ ′ ′ ′ ′ Fi = Fi −e+f has this property. Let F = F −Fi +Fi , D = D−e and A = A−f. Then F ′,A′,D′ satisfy condition (P1), (P2) and (P3).

Case 2.2: The subdivision is improper. Suppose that f is a loop. It is worth keeping in mind that g and e have the same ends. ′ ′ ′ ′ If σg = σe then we just put Fi = Fi − e + g, F = F − Fi + Fi A = A − g and D′ = D − e. The conditions (P1), (P2), (P3) obviously hold for F ′,A′,D′. So let us suppose that σg > σe and hence σf = σe. ′ ′ If σe ≥ ti or there is no cycle containing e in F , set Fi = Fi − e + g, A = A − g ′ ′ and D = D − e. Then Fi is a pseudoforest and condition (P2), (P3) obviously ′ ti hold. To check condition (P1’), suppose by contradiction that (Fi ) contains cycle ′ ti C . Then the cycle must contain g since Fi has no cycle. We obtain a cycle C in ′ Fi from C by replacing g with e. However, we have assumed that whether there is no cycle containing e in Fi or ti ≤ σe < σg, so we must have ti ≤ σe < σg. ti However, under this condition, C is a cycle in Fi , a contradiction. It means that ′ ′ ′ ′ (P1’) holds for Fi and so F ,A ,D satisfy (P1), (P2) and (P3).

Now consider the case when σe < ti and there is a cycle containing e in Fi. Put ′ ′ ′ ′ ′ Fi = Fi − e + f, F = F − Fi + Fi , A = A − f and D = D − e. Since there is ′ ′ a cycle containing e in Fi, Fi is a pseudoforest. Since the only new cycle in Fi is ′ ti that consists of the loop f, which is of grade σf = σe < ti,(Fi ) contains no cycle, ′ ′ ′ ′ so (P1’) holds for Fi . Therefore F ,A ,D satisfy (P1), (P2) and (P3). 2

80 Chapter 5. Inductive constructions and decompositions

5.2.5 Graded sparse matroids

This section is devoted to the study of (m, d)-graded sparse matroids, i.e. matroids whose independent sets are the edge sets of (m, d)-graded sparse graphs. The fact that these sets define a matroid was shown by Lee, Streinu and Theran [79] using the matroid circuit axioms. We will provide submodular functions that induce the (m, d)-graded sparse matroids. This gives an alternative proof for the result of [79], and also determines the rank formula for graded sparse matroids. We then use this rank formula to express an (m, d)-graded sparse matroid as a union of m smaller graded sparse matroids. This matroid decomposition results in a simple proof for Theorem 5.2.3. We first describe a general setting for an (m, d)-graded sparse matroid. We think of the ground set as a subset of the edge set of an r-graded graph on n r,m vertices, Kn , in which every vertex is adjacent to m loops of each grade, every pair of vertices are joined by 2m parallel edges of each grade, and n is a large r,m unspecified integer. Since we work with subsets E of the edges of Kn , it is convenient to reformulate the sparsity condition for such an edge set. We say that E is (m, d)-graded sparse if it is the edge set of an (m, d)-graded-sparse subgraph r,m of Kn . It is not difficult to see that an edge set E is (m, d)-graded sparse if and t t only if for every subset F of E, |F | ≤ m|V (F )| − dt for all 1 ≤ t ≤ r, where F is the set of all edges of F which have grade at least t and V (F ) is the set of all vertices which are incident with edges in F . r,m For a subset F of E(Kn ), let

fm,d(F ) = m|V (F )| − dt, (5.2.10) where t is the minimum grade of the elements in F when F =6 ∅, and put fm,d(∅) = 0. It is easy to see that fm,d is nonnegative and nondecreasing for d = (d1, d2, . . . , dr) when

0 ≤ d1 ≤ d2 ≤ · · · ≤ dr ≤ 2m − 1 and we will assume henceforth that this is the case. We also have the submodularity property.

Lemma 5.2.16. The function fm,d is an intersecting submodular function on the r,m subsets of E(Kn ).

Proof. Let F1,F2 be edge sets with F1 ∩ F2 =6 ∅. Let t1, t2 be the minimum grades of edges in F1,F2 respectively. We can suppose without loss of generality that

81 5.2. Graded sparse graphs

t1 ≤ t2. Then t1 is the minimum grade of edges in F1 ∪F2, and the minimum grade of edges in F1 ∩ F2 is at least t2 . We have,

fm,d(F1) + fm,d(F2) = m|V (F1)| − dt1 + m|V (F2)| − dt2

≥ m|V (F1 ∪ F2)| − dt1 + m|V (F1 ∩ F2)| − dt2

≥ fm,d(F1 ∪ F2) + fm,d(F1 ∩ F2) where the ﬁrst inequality uses the fact that the function F 7→ |V (F )| is submodular.

r,m Theorem 2.4.2 now implies that fm,d induces a matroid on E(Kn ). We denote this matroid by M(m, d). We next show that the independent sets of M(m, d) are the (m, d)-graded sparse edge-sets.

Theorem 5.2.17. The edge sets of all (m, d)-graded sparse graphs form the independent sets of the matroid M(m, d).

Proof. Suppose that E is the edge set of an (m, d)-graded sparse graph and ∅= 6 F ⊆ E. Let t be the minimum grade of an edge in F . Then |F | = |F t| ≤ m|V (F )| − dt = fm,d(F ). Thus E is independent in M(m, d). Conversely, suppose that E is independent in M(m, d). We show that the graph (V (E),E) is (m, d)-graded sparse. Let ∅= 6 F ⊆ E and let t be the minimum ′ t′ t grade of an edge in F . Then fm,d(F ) = m|V (F )| − dt. For t ≤ t, |F | = |F | ≤ fm,d(F ) = m|V (F )| − dt ≤ m|V (F )| − dt′ , by the (m, d)-sparseness of F and the ′ t′ t′ definition of fm,d. For t ≥ t, |F | ≤ fm,d(F ) ≤ m|V (F )| − dt′ , where the first inequality follows from the fact that F is independent in M(m, d) and the second inequality follows from the definition of fm,d, since the minimum grade of an edge ′ in F t is at least t′ and the sequence d is nondecreasing. Therefore, (V (E),E) is indeed (m, d)-graded sparse. The theorem follows.

We will refer to M(m, d) as the (m, d)-graded sparse matroid. Theorem 2.4.2 and Theorem 5.2.17 determine the rank formula for M(m, d).

Corollary 5.2.18. The rank function of M(m, d) is given by

rM(m,d)(E) = min fm,d(Fj) + |F0| : {F0,F1 ...,Fs} partitions E . j=1 X In the remainder of this section we assume that d satisﬁes the stronger hypothesis that

0 ≤ d1 ≤ · · · ≤ dr ≤ m.

82 Chapter 5. Inductive constructions and decompositions

i i i For 1 ≤ i ≤ m, we deﬁne the r-tuple c = (c1, . . . , cr) by

i 1 if i ≤ dt, ct = ( 0 otherwise.

m i Note that we have dt = i=1 ct, for all t = 1, . . . , r. We will show that the (m, d)-sparse matroid is the union of the m (1, ci)-graded sparse matroids, i.e. P M(m, d) = M(1, c1) ∨ M(1, c2) ∨ · · · ∨ M(1, cm).

We will need the following result.

Lemma 5.2.19. Suppose H = (V,F ) is a connected graph. If H is (m, d)-graded sparse and |F | > m|V | − dt+1 for some 1 ≤ t ≤ r then there exists an edge e belonging to a cycle of H with σe ≤ t.

Proof. First note that since H is (m, d)-graded sparse, |F | ≤ m|V |, and hence we t+1 must have dt+1 ≥ 1. We also have |F | ≤ m|V | − dt+1 from the sparseness of H.

Since |F | > m|V | − dt+1, there exists an edge in F of grade at most t. Suppose that every edge of grade at most t in F is a cut-edge of H. Let F1,...,Fs be the edge sets of the connected components of the subgraph of H induced by F t+1 and s F0 = F \ i=1 Fi. Since H is connected and all edges in F0 are cut-edges, we have |F | = s − 1. Thus 0 S s

|F | = |F0| + |Fi| i=1 Xs

≤ |F0| + (m|V (Fi)| − dt+1) i=1 X = (s − 1) + m|V | − dt+1 − (s − 1)dt+1

≤ m|V | − dt+1.

This contradiction to the assumption that |F | > m|V | − dt+1 implies that there is an edge e with σe ≤ t that belongs to a cycle of H.

Now we are ready to prove the following main result of this section.

Theorem 5.2.20. Suppose 0 ≤ d1 ≤ · · · ≤ dr ≤ m. Then the matroid M(m, d) is the union of the m matroids M(1, ci) for i = 1, . . . , m.

Proof. By deﬁnition

i t i if Ii is independent in M(1, c ) then |Ii | ≤ |V (Ii)| − ct (5.2.11)

83 5.2. Graded sparse graphs for all t = 1, . . . , m. Let M = M(1, c1) ∨ · · · ∨ M(1, cm). We need to show that M = M(m, d).

First suppose that I is an independent set of M. Then I = I1 ∪ · · · ∪ Im where i Ii is an independent set in M(1, c ) for each i ∈ {1, . . . , m}, by the deﬁnition of matroid union. Then for every t, 1 ≤ t ≤ r, and for every i = 1, . . . , m we have, t i |Ii | ≤ |V (Ii)| − ct by (5.2.11). Therefore,

m m t t i |I | = |Ii | ≤ (|V (Ii)| − ct) ≤ m|V (I)| − dt, i=1 i=1 X X for all t = 1, . . . , r. We have a similar inequality for every subset J of I. Hence (V (I),I) is indeed (m, d)-graded sparse, and hence I is independent in M(m, d) by Theorem 5.2.17. Conversely, suppose that I is an independent set of M(m, d). We will show that I is also independent in M. We do this by showing that the rank of I in M is equal to |I|.

By Theorem 2.3.1, to prove that rM(I) = |I| it is suﬃcient to show that

rM(1,c1)(F ) + ··· + rM(1,cm)(F ) ≥ |F | holds for every subset F of I.

Let F1,...,Fk be the edge sets of the connected components of the subgraph H r,m of Kn induced by F . Then |F | = |F1|+···+|Fk| and rM(1,ci)(F ) = rM(1,ci)(F1)+

··· + rM(1,ci)(Fk) for all i = 1, . . . m. Hence we may suppose that H is connected. Let t∗ be minimum grade of an edge e which belongs to a cycle C of H, where we set ∗ t = 0 if no such edge exists. Then Lemma 5.2.19 implies that |F | ≤ m|V (F )|−dt∗ , ∗ taking dt∗ = 0 when t = 0. On the other hand, it is not diﬃcult to see that

|V (F )| − 1, if 1 ≤ i ≤ dt∗ , rM(1,ci)(F ) = ( |V (F )|, otherwise.

Since F is independent in M(m, d), this gives

rM(1,c1)(F ) + ··· + rM(1,cm)(F ) = m|V (F )| − dt∗ ≥ |F |.

The theorem now follows.

An immediate consequence of this result is that the union of two graded sparse matroids is again a graded sparse matroid.

84 Chapter 5. Inductive constructions and decompositions

′ ′ ′ Corollary 5.2.21. Let d = (d1, . . . , dr), d = (d1, . . . , dr) be two r-tuples of in- ′ tegers and m, m be positive integers such that 0 ≤ d1 ≤ · · · ≤ dr ≤ m and ′ ′ ′ 0 ≤ d1 ≤ · · · ≤ dr ≤ m . Then the union of two graded sparse matroids M(m, d) and M(m′, d′) is the graded sparse matroid M(m + m′, d + d′), where d + d′ = ′ ′ (d1 + d1, . . . , dr + dr), i.e.,

M(m + m′, d + d′) = M(m, d) ∨ M(m′, d′).

The decomposition of graded sparse graphs is also a consequence of this matroid decomposition result.

Proof of Theorem 5.2.3 from matroid decomposition

Let m, d and ci, i = 1, . . . , m, be defined as above. It is not difficult to see that a i base Fi of M(1, c ) induces a pseudoforest such that the restriction of Fi on Et is i i a forest if and only if ct = 1. However, by the definition of c , this is equivalent to the condition that the restriction of Fi on Et is a forest if and only if 0 ≤ i ≤ dt. i Moreover, Fi is a spanning forest if and only if c1 > 0 and Fi is a tight spanning i pseudoforest if and only if c1 = 0. Therefore, the matroid decomposition in Theo- rem 5.2.17 induces our desired graph decomposition. 2

85 5.3. (b, l)-sparse graphs

5.3 (b, l)-sparse graphs

Let V be a ﬁnite set, b : V → Z+, we use bmin to denote min {b(v): v ∈ V }. In this section we suppose that l is an integer satisfying

0 ≤ l < 2bmin.

Recall from Chapter 2 that a graph G = (V,E) on V is said to be (b, l)-sparse if i(X) ≤ b(X) − l for every X ⊆ V with i(X) > 0. Contribution: This section provides an inductive construction of (b, l)-sparse graphs for 0 ≤ l ≤ bmin. We also extend the pebble game in [78] to characterize

(b, l)-sparse graphs for 0 ≤ l < 2bmin.

5.3.1 Inductive construction of (b, l)-sparse graphs, 0 ≤ l ≤

bmin

Throughout this subsection we suppose that 0 ≤ l ≤ bmin. The sparsity condition is then written simply as i(X) ≤ b(X)−l for every ∅= 6 X ⊆ V . Due to the matroidal property of the family of (b, l)-sparse graphs, a (b, l)-sparse graph can always be obtained from a (b, l)-tight graph by deleting some edges. So in this subsection we are interested only in the inductive construction of (b, l)-tight sparse graphs. The proof of our inductive construction for (b, l)-tight sparse graphs G = (V,E) is proceeded by induction on the number of vertices and follows the same line as that for (m, d)-graded tight graphs: ﬁrst we choose a vertex v ∈ V with some requirement on the number of incident edges, then we show that for this vertex there is a reduction that results in a (b, l)-tight graph on V − v. This proof is relatively simpler than the one for (m, d)- tight graphs but we include it for the sake of completeness. Also, since situations in the two proofs are quite similar, we will abuse several terms already used in the previous section. The required condition for the chosen vertex v is that d(v) = b(v) + k + i(v) with 0 ≤ k + i(v) ≤ b(v). The existence of a vertex satisfying this condition is assured by the following lemma. Lemma 5.3.1. Let G = (V,E) be a (b, l)-tight sparse graph on at least two vertices. Then there exists a vertex v ∈ V such that d(v) = b(v) + k + i(v) for some k ≥ 0 such that 0 ≤ k + i(v) ≤ b(v). Proof. First, for every v ∈ V , we have d(v) − i(v) = i(V ) − i(V − v) ≥ b(V ) − l − (b(V − v) − l) = b(v).

86 Chapter 5. Inductive constructions and decompositions

Suppose by contradiction that there is no vertex v ∈ V satisfying the condition in the lemma, then for every v ∈ V , d(v) > 2b(v) holds. Then

2|E| = d(v) > 2 b(v) = 2b(V ) = 2(|E| + l), v∈V v∈V X X a contradiction. The lemma follows.

A reduction operation consists of splitting offs and loop deletions. A splitting off at a vertex v of a graph G is an operation that deletes two edge f = vx, g = vy incident to v and adds an edge e = xy. If f is a loop this operation simply deletes f, we refer to this as a loop deletion. If neither f nor g is a loop, we call the operation a proper splitting off. Let v be a vertex of a (b, l)-tight graph G incident with c loops and d(v) = b(v) + k + c, for some 0 ≤ k ≤ m.A k-reduction at v first does c loop deletions at v following by k proper splitting offs at v, then deletes all the remaining b(v) − k edges. Note that when k = c = 0, the k-reduction simply deletes from G the vertex v and all its incident edges. A k-reduction at v is admissible if the resulting graph is also (b, l)-sparse. Suppose that H is a (b, l)-sparse graph and v is a vertex of H with

d(v) = b(v) + k + c − 2s, i(v) = max {0, c − s}, (5.3.1) c ≥ 0, k ≥ 1 and 0 ≤ s < k + c ≤ b(v). We can imagine that c is the number of loops at v in G and H is obtained from G after s splitting oﬀs. This condition implies that

The number of edges incident to v is strictly greater than b(v) − s. (5.3.2)

In fact, this number is d(v) − i(v) = b(v) + k + c − 2s − i(v). If i(v) = 0 then b(v) + k + c − 2s − i(v) = b(v) + k + c − 2s > b(v) − s since s < k + c by assumption 5.3.1. If i(v) > 0 then i(v) = c − s by assumption 5.3.1. Hence b(v) + k + c − 2s − i(v) = b(v) + k + c − 2s − (c − s) = b(v) + k − s > b(v) − s since k ≥ 1 by assumption 5.3.1. Assumption 5.3.1 also implies that b(v) − i(v) ≥ b(v) − (c − s) ≥ k + s ≥ 1. Therefore, we have there are non loop edges at v. (5.3.3) We also say that H is (v, s)-good if i(X) ≤ b(X) − l for every X ⊆ V and i(X) ≤ b(X) − l − s for every X ⊆ V that properly contains v. In the rest of this section we will assume that H is (v, s)-good. (5.3.4)

87 5.3. (b, l)-sparse graphs

We will show that we can find a splitting off at v that results in a (v, s + 1)-good graph. Such a splitting off is called feasible. The two potential obstacles for a feasible splitting off are critical sets and crucial sets. A critical set is a set X ⊆ V − v with i(X) = b(X) − l.A crucial set is a set {v} ( X ⊂ V with i(X) = b(X) − l − s. More precisely, we have the following.

Lemma 5.3.2. A splitting oﬀ of f = vx, g = vy at v is feasible if and only if

1. no critical set contains both x and y, and 2. all crucial sets intersect the set {x, y}.

Proof. Let H′ be the graph obtained from H by splitting oﬀ f, g at v. First suppose that the splitting oﬀ of f, g at v is feasible, i.e., H′ is (v, s + 1)-good. Then for every critical set X ⊆ V − v, b(X) − l = iH (X) ≤ iH′ (X) ≤ b(X) − l. Hence, equality must holds everywhere, in particular, iH (X) = iH′ (X), which implies that X does not contain both x and y. Every crucial set Y must intersect {x, y} since, otherwise, we have b(Y )−l−s−1 ≥ iH′ (Y ) = iH (Y ) = b(Y )−l−s, a contradiction. Conversely, suppose that there is no critical set containing both x and y and all crucial sets intersect {x, y}. Let X be a subset of V − v. If X is not a critical set then iH′ (X) ≤ iH (X) + 1 ≤ b(X) − l holds. If X is a critical set then X does not contain both x and y, thus iH′ (X) = iH (X) = b(X) − l holds. Now let

Y be a subset of V that properly contains v. If Y is not crucial then iH′ (Y ) ≤ iH (Y ) ≤ b(Y ) − l − s − 1 holds. If Y is a crucial set then Y intersects {x, y}, so iH′ (Y ) ≤ iH (Y ) − 1 ≤ b(Y ) − l − s − 1 holds. Therefore, splitting oﬀ f, g at v is feasible.

In the following, we will investigate properties of critical sets and crucial sets. The following relations will be often of use.

i(X) + i(Y ) + i(X \ Y,Y \ X) = i(X ∪ Y ) + i(X ∩ (Y ).

In particular, i(X) + i(Y ) ≤ i(X ∪ Y ) + i(X ∩ (Y ).

It is also useful keeping in mind that

b(X) + b(Y ) = b(X ∪ Y ) + b(X ∩ Y ).

The ﬁrst result shows that the intersection and union of two intersecting critical sets are critical sets.

88 Chapter 5. Inductive constructions and decompositions

Lemma 5.3.3. Let X,Y be two critical sets with X ∩ Y =6 ∅. Then X ∪ Y and X ∩ Y are critical sets. Moreover, i(X \ Y,Y \ X) = 0. Proof. We have

b(X) − l + b(Y ) − l + i(X \ Y,Y \ X) = i(X) + i(Y ) + i(X \ Y,Y \ X) = i(X ∪ Y ) + i(X ∩ Y ) ≤ b(X ∪ Y ) − l + b(X ∩ Y ) − l = b(X) − l + b(Y ) − l.

Therefore equality must hold everywhere, so i(X ∪ Y ) = b(X ∪ Y ) − l, i(X ∩ Y ) = b(X ∩ Y ) − l and i(X \ Y,Y \ X) = 0. The lemma follows.

An immediate corollary of this lemma is the following. Corollary 5.3.4. The union of a collection of critical sets containing a common element x is a critical set. 2 The next result indicates that the number of edges from a critical set to v is upperbounded by b(v) − s − i(v). Lemma 5.3.5. Let X be a critical set. Then i(v, X + v) ≤ b(v) − s. Proof. Since X is critical we have, b(X)+b(v)−l−s = b(X +v)−l−s ≥ i(X +v) = i(X) + i(v, X + v) = b(X) − l + i(v, X + v). Hence i(v, X + v) ≤ b(v) − s.

The following result shows that if two crucial sets intersect in at least two vertices then their union and intersection are also crucial. Lemma 5.3.6. Let X,Y be crucial sets with |X ∩ Y | ≥ 2, then X ∪ Y and X ∩ Y are crucial. Proof. The proof is similar to that about critical sets. We include it for the sake of completeness.

b(X) − l − s + b(Y ) − l − s ≤ i(X) + i(Y ) ≤ i(X ∪ Y ) + i(X ∩ Y ) ≤ b(X ∪ Y ) − l − s + b(X ∩ Y ) − l − s = b(X) − l − s + b(Y ) − l − s.

Hence equality must hold everywhere, so i(X ∪Y ) = b(X ∪Y )−l−s and i(X ∩Y ) = b(X ∩ Y ) − l − s. The lemma follows.

2 In fact, the following stronger assertion holds: Let X1,...,Xt be critical sets such that

Xi ∩ ( 1≤j≤i−1 Xj ) 6= ∅ for every i = 1, . . . , t. Then 1≤i≤t Xi is a critical set. S S 89 5.3. (b, l)-sparse graphs

In contrast to critical sets, the number of edges from a crucial set to v is lowerbounded by b(v) − s − i(v) as showed in the following result.

Lemma 5.3.7. Let X be a crucial set. Then i(v, X) ≥ b(v) − s. Moreover, i(v, X − v) ≥ 1.

Proof. We have

b(X) − l − s = i(X) = i(X − v) + i(v, X) ≤ b(X − v) − l + i(v, X) = b(X) − b(v) − l + i(v, X).

Therefore i(v, X) ≥ b(v) − s. Now suppose by contradiction that i(v, X − v) = 0. Then i(v) = i(v, X) ≥ b(v) − s > 0 by assumption 5.3.1. Therefore, again by assumption 5.3.1, i(v) = c − s. Hence we have, c − s ≥ b(v) − s, which implies c ≥ b(v), a contradiction to the assumption b(v) ≥ k + c and k ≥ 1.

The next result shows that if two crucial sets intersect at only v, then their union contains all the neighbors of v.

Lemma 5.3.8. Let X,Y be crucial sets with X ∩ Y = {v}. Then E(v, V ) = E(v, X ∪ Y ).

Proof. We have

b(v) + k + c − 2s = d(v) (by assumption 5.3.1) ≥ i(v, X) + i(v, Y ) (since X ∩ Y = {v}) = b(v) − s + b(v) − s (by Lemma 5.3.7) ≥ b(v) − 2s + k + c (by assumption 5.3.1)

Therefore equality must hold everywhere, in particular, d(v) = i(v, X) + i(v, Y ). Thus E(v, V ) = E(v, X ∪ Y ).

We will also need the following result.

Lemma 5.3.9. There are at most two diﬀerent (inclusionwise) minimal crucial sets in H.

90 Chapter 5. Inductive constructions and decompositions

Proof. Suppose that X,Y,Z are three minimal crucial sets in H. If |X ∩ Y | ≥ 2 then, by Lemma 5.3.6, X ∩ Y is a crucial set, so X must coincide to Y by the minimality. So let us suppose that X ∩ Y = {v}. Lemma 5.3.8 implies that X ∪ Y contains all the neighbors of v. By Lemma 5.3.7, Z − v contains a neighbor z of v, so z must belong to X ∪ Y . Assume without loss of generality that z ∈ X. Then |X ∩ Z| ≥ 2, thus X ∩ Z is a crucial set, so X = Z by the minimality. The lemma follows.

Lemma 5.3.10. There exists a feasible splitting oﬀ at v.

Proof. Since all crucial sets contain v, a loop deletion at v is alway feasible by Lemma 5.3.2. So if there exists a loop at v, the lemma holds. Therefore we may suppose that i(v) = 0. Let us first consider the case that there exist two different minimal crucial sets X and Y . Then let f = vx be an edge in E(v, X − v) and g = vy be an edge in E(v, Y − v) (their existence is guaranteed by Lemma 5.3.7). Our aim is to prove that splitting off f, g at v is feasible. Since every crucial set contains either X or Y by Lemma 5.3.9, the second condition on crucial sets of Lemma 5.3.2 is sastisfied. It remains to show that there is no critical set containing both x and y. Suppose on the contrary that U is a critical set containing both x and y. A similar proof to that of Lemma 5.3.3 shows that i(U \ X,X \ U) = 0, contradicting the fact that g = vy ∈ E(U \ X,X \ U). Therefore, both conditions in Lemma 5.3.2 is satisfied, so splitting off f, g at v is feasible. Now let us consider the case that there exists at most one minimal crucial set. If one exists, let X denote the minimal crucial set, otherwise, let X = V . Let f = vx be an edge in E(v, X − v) by Lemma 5.3.7. If there exists a critical set containing x, let U be the maximal one (U is in fact the inclusionwise maximum critical set containing x by Corollary 5.3.4). Then by Lemma 5.3.5, i(v, U + v) ≤ b(v) − s while i(v, V ) > b(v) − s by (5.3.2). Therefore, there exists an edge g = vy ∈ E(v, V − v − U). Splitting off f, g at v is then feasible by Lemma 5.3.2.

Now we are ready to prove the main reduction lemma.

Lemma 5.3.11 (Reduction lemma). If G = (V,E) is a (b, l)-tight graph then there exists an admissible k-reduction at a vertex v of G.

Proof. We choose a vertex v ∈ V incident with c loops and d(v) = b(v) + k + c for some 0 ≤ k ≤ b(v) which exists by Lemma 5.3.1. If k = 0 we simply delete

91 5.3. (b, l)-sparse graphs the vertex v. The obtained graph H is obviously (b, l)-sparse and it is tight as well since |E(H)| = |E(G)| − (d(v) − i(v)) = b(V ) − l − b(v) = b(V (H)) − l. If k ≥ 1, we initialize the induction steps with s = 0. Lemma 5.3.2 shows that, as long as s < k + c, we can find a splitting off at v such that the obtained graph is (v, s + 1)-good. Therefore, we can do k + c feasible splitting offs and then delete the vertex v. The obtained graph H is (b, l)-sparse by the definition of feasible splitting offs. It remains to check that H is tight. Using the tightness of G we have |E(H)| = |E(G)| − c − k − (b(v) − k) = b(V ) − l − c − k − (b(v) − k) = b(V (H)) − l. The lemma follows.

Next we deﬁne the reverse operation of k-reduction, that we call k-extension.

Let b : V → Z+ an integer-valued function on a set V .A k-extension on a (b, l)-sparse graph H with V (H) ⊂ V is an operation that

1. chooses a vertex v ∈ V − V (H);

2. delete k ≤ b(v) edges ei = xiyi, i = 1, . . . k from H;

3. for each i = 1, . . . , k, add edges vxi, vyi to H; then 4. adds b(v) − k edges incident with v such that there are at most b(v) − l loops among them.

We call this operation a k-extension on e1, . . . , ek with new vertex v. Note that the number of edges increases by b(v). The main theorem of this section characterizes (b, l)-tight graphs in term of inductive construction.

Theorem 5.3.12. Let b : V → Z+ and 0 ≤ l ≤ min {b(u): u ∈ V }. A graph G = (V,E) on V is (b, l)-tight if and only if it can be constructed from a graph on a single vertex v0 ∈ V with b(v0) − l loops by a sequence of k-extensions.

Proof. The “if” part follows from Lemma 5.3.11 and by induction. We prove the

“only if” part by induction. The graph on a single vertex v0 with b(v0) − l loops is clearly (b, l)-tight. Suppose that a graph G = (V,E) is obtained from a (b, l)-tight graph H by a k-extension on k edges ei = xiyi for i = 1, . . . , k with the vertex v ∈ V − V (H). We check the (b, l)-tightness of G. For a subset X of V − v, iG(X) ≤ iH (X) ≤ b(X) − l by the sparseness of H. If X = {v}, iG(v) ≤ b(v) − l by the condition on the number of added loops. For a subset X of V that contains v, if X contains both ends xi, yi of an edge ei, then when comparing to EH (X), in EG(X) two edges vxi, vyi are added while an edge ei is deleted. So the total

92 Chapter 5. Inductive constructions and decompositions

contribution of the deletion of ei and addition of vxi, vyi to EG(X) is at most one. The contribution of any other added edge to EG(X) is also at most one.

Therefore, iG(X) ≤ iH (X − v) + b(v) ≤ b(X − v) − l + b(v) = b(X) − l holds. Hence G is (b, l)-sparse. Furthermore, since H is tight, |E(G)| = |E(H)| + b(v) = b(V (H)) − l + b(v) = b(V (G)) − l. The theorem follows.

5.3.2 (b, l)-pebble games

In this subsection, we introduce (b, l)-pebble games which generalizes (m, l)-pebble games by Lee and Streinu [78]. We show that (b, l)-sparse graphs can be characterized using (b, l)-pebble games.

Let V be a ﬁnite set of vertices, b : V → Z+ and 0 ≤ l ≤ 2bmin. Let G = (V,E) be an undirected graph on V . In a (b, l)-pebble game, each vertex v of V is provided with b(v) pebbles. We consider an unoriented edge uv of G and try to orient it using the following rules.

1. Add-edge move: If there are totally at least l + 1 pebbles at u and v and p is a pebble at u, then put p on uv and orient uv from u to v. We say that uv is covered by p.

2. If there are totally at most l pebbles at u and v, then try to collect at least l + 1 pebbles at u and v using pebble-slide moves which will be described later.

3. If it is impossible to collect at least l + 1 pebbles at u and v then discard the edge uv.

A pebble-slide move concerns an oriented edge uv (oriented from u to v). Let p be the pebble placed on uv and p′ a pebble at v. A pebble-slide move puts p back to u, places p′ on uv and reorients uv from v to u. Note that in a pebble game, a pebble stays either in its original vertex or on an edge adjacent to this vertex and once an edge is oriented, it remains an oriented edge (althought may be reoriented to the inverse direction). At the end of the game, we obtain a subgraph H of G whose edges are all the edges of G that are oriented by the pebble game. We say that H is constructed by a (b, l)-pebble game. The (b, l)-sparseness of H follows from the following lemma. At some stage in the pebble game, for a subset U of V , let peb(U) denote the total number of pebbles at all the vertices of U, span(U) denote the number of oriented edges with both ends in U and out(U) denote the number of oriented edges with tail in U and head in V \ U. If U = {v}, we simply

93 5.3. (b, l)-sparse graphs use peb(v), span(v) and out(v) to refer to peb(U), span(U) and out(U). We denote by D the directed graph obtained at stage of a game.

Lemma 5.3.13. During the pebble game, the following conditions always hold on D for every v ∈ V and for every U ⊆ V .

1. peb(v) + span(v) + out(v) = b(v);

2. peb(U) + span(U) + out(U) = b(U);

3. peb(U) + out(U) ≥ l if l ≤ bmin or |U| ≥ 2;

4. span(U) ≤ b(U) − l if l ≤ bmin or |U| ≥ 2.

Proof. The first two equalities follows from the fact that in a pebble game, a pebble always stays either in its original vertex or on an edge oriented out from this vertex. Hence peb(v) + span(v) + out(v) is the total number of pebbles originally placed at v and so is equal to b(v). Similarly, peb(U) + span(U) + out(U) is the total number of pebbles originally placed at the vertices of U and so is equal to b(U). The last inequality follows easily from the second equality and the third inequality. Now we prove the third inequality by induction on the number of moves. At the beginning, all the pebbles are placed at the vertices, so peb(U) + out(U) = b(U) ≥ l if l ≤ bmin or |U| ≥ 2. It is sufficient to show that after each move, this condition always holds. First consider an add-edge move that orients an unoriented edge uv from u to v. Then by definition, before the move pev(u) + peb(v) ≥ l + 1, so after the move pev(u)+peb(v) ≥ l holds. Hence if both u, v are in U we have peb(U)+out(U) ≥ l. If at most one of u, v belongs to U then this move does not decrease out(U), and if peb(U) is decreased by this move it is because u ∈ U, v∈ / U and one pebble from u is put on uv, which decreases peb(U) by one. However, in this case, the move increases out(U) by one. Therefore the condition holds after an add-edge move. Now consider a pebble-slide move on an oriented edge uv (oriented from u to v.) If both u, v are in U or both u, v are not in U then the move does not change peb(U) + out(U), so the condition holds after this pebble-slide move. If u ∈ U and v ∈ V \ U then the move decreases out(U) by one in reorienting uv, but it increases peb(U) by one by pushing back a pebble on uv to u. So in this case, peb(U) + out(U) remains the same. If u ∈ V − U and v ∈ U, the move decreases peb(U) by one but increases out(U) by one. So peb(U) + out(U) remains the same in this case too. Therefore, the condition holds after a pebble-slide move. The lemma follows.

94 Chapter 5. Inductive constructions and decompositions

An immediate consequence of Lemma 5.3.13 is the following.

Corollary 5.3.14. The resulting graph in (b, l)-pebble game is a (b, l)-sparse graph.

The next result shows that if we play the (b, l)-pebble game on a (b, l)-sparse graph G then no edge is discarded.

Lemma 5.3.15. Let H be the underlying graph of the directed graph obtained at a stage of the (b, l)-pebble game on a graph G. Suppose that an unoriented edge uv is then considered. Then uv is discarded by the pebble game if and only if uv is in the closure of E(H) in the (b, l)-count matroid on G.

Proof. If uv is not discarded by the (b, l)-pebble game then it is added to H. so Corollary 5.3.14 implies that H + uv is (b, l)-sparse. Therefore uv is not in the closure of E(H) in the (b, l)-count matroid on G. Now suppose that uv is discarded by the pebble game. Let D be the directed graph obtained at the stage just before we discard uv and after all the possible pebble-slide moves to collect pebbles at u and v. Let Reach(u) and Reach(v) denote the set of vertices reachable from u, v, respectively, in D and let X = Reach(u) ∪ Reach(v). Since no more pebble can be collected to u and v, there is no pebble on every vertex x of X other than u, v, otherwise, we can do pebble-slide moves on the dipath from, say, u to x to collect one more pebble at u and v. Hence peb(X) = peb(u) + peb(v) ≤ l. Moreover, out(X) = 0. Therefore, condition 2 of

Lemma 5.3.13 implies that iH (X) = span(X) = b(X)−peb(u)−peb(v) ≥ b(X)−l.

Since H is (b, l)-sparse, iH (X) = b(X) −l holds. Hence H + uv is not (b, l)-sparse, so uv is in the closure of E(H) in the (b, l)-count matroid on G.

Therefore we obtained the following characterization of (b, l)-sparse graphs in term of pebble games.

Theorem 5.3.16. A graph is (b, l)-sparse if and only if it is obtained by a (b, l)- pebble game.

95 5.4. Packing of matroid-based arborescences

5.4 Packing of matroid-based arborescences

5.4.1 Introduction

Recall from Section 2.2 that an out-arborescence (resp., in-arborescence) is a directed graph in which each vertex has in-degree (resp., out-degree) at most 1 and there is exactly one vertex r with in-degree (resp., out degree) 0. Namely, an out-aborescence is a tree where all edges are oriented away from a root node while an in-arborescence is a tree where all edges are oriented toward a root node. The problem of packing arborescences has important applications in many practical problems such as evacuation, commodity, broadcasting,. . . For example, in evacuation situation (tsunami, earthquake, ﬁre,. . . ), an in-arborescence represents roads used by refugees while an out-arborescence can represent the roads use by emer- gency vehicles. Since each road has a limited capacity, it is preferable to have several disjoint paths from each place to the safe place, where comes the necessity of considering packing of arborescences. In broadcast networks or commodity networks, informations or commodities are also sent along arcs of arborescences. The existence of a packing of arborescences satisfying some required conditions makes sure that the informations or commodities are sent without interference. From a theoretical view point, the following two problems are fundamental.

1. Given a digraph (a network with pre-determined orientation of edges), discern whether there exists a certain number of arc-disjoint arborescences satifying some conditions.

2. Given an undirected graph, discern whether one can orient the edges of the graph such that in the oriented graph there exists a certain number of arc- disjoint arborescences satisfying some conditions. This problem reduces to the problem of discerning whether there exists a certain number of edge- disjoint trees satisfying the corresponding conditions.

The earliest fundamental results on these problems are due to Nash-Williams, Tutte and Edmonds.

Theorem 5.4.1 (Nash-Williams [82], Tutte [104]). A graph G = (V,E) contains k edge-disjoint spanning trees if and only if the inequality

eG(P) ≥ k|P| − k holds for every partition P of V .

96 Chapter 5. Inductive constructions and decompositions

Theorem 5.4.1 is the dual of Theorem 5.1.1

Theorem 5.4.2 (Edmonds [31]). Let D = (V,A) a digraph and r ∈ V . The digraph D contains k arc-disjoint spanning out-arborescences rooted at r if and only if

ρD(X) ≥ k for every nonemty X ⊆ V − r.

Frank pointed out that the undirected result of Nash-Williams and Tutte can be obtained easily from its directed counterpart of Edmonds through an orientation result.

Theorem 5.4.3 (Frank [39]). Let G = (V,E) be an undirected graph and r a vertex of G. There exists an orientation D of G such that ρD(X) ≥ k for every nonempty X ⊆ V − r if and only if eG(P) ≥ k|P| − k for every partition P of V .

The above result of Edmonds is generalized in many ways, among them we can name results on packing of branchings, packing of Steiner arborescences3. One of the notable generalizations is the following by Kamiyama, Katoh and Takizawa. For a digraph D = (V,A) and a vertex r ∈ V , let Reach(r) denote the set of all vertices v reachable from r, i.e., there exists a directed path from r to v in D. In the remainder of this section we will simply use “arborescence” to refer to “out-arborescence”.

Theorem 5.4.4 (Kamiyama, Katoh and Takizawa [71]). Let D = (V,A) be a digraph and R = {r1, . . . , rt} a multiset of elements in V . There exist k arc-disjoint arborescences T1,...,Tk in D such that Ti is rooted at ri and spans Reach(ri) if and only if

ρD(X) ≥ p(X) for every nonempty X ⊂ V, where p(X) denotes the number of i ∈ {1, . . . , t} such that ri ∈/ X and there exists a directed path from ri to an element in X.

Fujishige generalizes this results for packing arc-disjoint arborescences spanning convex sets. A convex set in a digraph D = (V,A) is a set U ⊆ V such that every dipath from u to v with u, v ∈ U lies completely in U. Note that, for each i ∈ {1, . . . , t}, Reach(ri) is obviously a convex set.

3The problem of discerning whether a directed graph contains k edge-disjoint Steiner arborescences is NP-hard in general, but it is polynomial when some condition on the in-degree of terminals set is satisﬁed [11]

97 5.4. Packing of matroid-based arborescences

Theorem 5.4.5 (Fujishige [41]). Let D = (V,A) be a digraph, R = {r1, . . . , rt} a multiset of elements of V and U1,...,Ut convex sets with ri ∈ Ui for i = 1, . . . , t.

There exist arc-disjoint arborescences T1,...,Tt rooted at r1, . . . , rt such that Ti spans Ui, for i = 1, . . . , t, if and only if

(U1,...,Uk) ρD(X) ≥ pR (X) for all nonempty X ⊂ V

(U1,...,Uk) where pR (X) denotes the number of i ∈ {1, . . . , t} such that Ui ∩ X =6 ∅ and ri ∈/ X.

We generalize the result of Edmonds in another direction. Suppose that S =

{s1,..., st} is a ﬁnite set and D = (V,A) is a digraph and π : S → V is a map. We can think of π as a placement of elements of S on vertices of D. An S-rooted arborescence of D is a pair (T, s) where T is an arborescence of D rooted at a vertex r ∈ V with π(s) = r. We will also call s the root of T , S the root set of (D, π) and we say that T is rooted at s. When S is clear from the context we simply call an S-rooted arborescence a rooted-arborescence. Let M be a matroid on S. We call the quadruple (D, M, S, π) a matroid-based rooted-digraph.A matroid-based packing of arborescences is a set {(T1, s1),..., (Tt, st)} of pairwise arc-disjoint S-rooted arborescences of D such that for each v ∈ V , the −1 set {si ∈ S : v ∈ V (Ti)} forms a base of M (Figure 5.1). We denote SX = π (X) for X ⊆ V and Sv = S{v} for v ∈ V . Our main result in this section is the following.

Theorem 5.4.6. A matroid-based rooted-digraph (D, M, S, π) has a matroid-based packing of arborescences if and only if the following conditions are satisﬁed.

1. Sv is independent for every v ∈ V .

2. For every nonempty subset X ⊂ V , the inequality ρD(X) ≥ rM(S) − rM(SX ) holds.

A map π satisfying condition 1 in Theorem 5.4.6 is said to be M-independent. A quadruple (D, M, S, π) satisfying condition 2 in this theorem is said to be rooted- connected. Our result is in fact motivated by the study of the rigidity of frameworks, or more precisely, by the work of Katoh and Tanigawa on matroid-based packing of trees, which takes it motivation from the study on inﬁnitesimal rigidity of frameworks with boundaries (see Section 6.2). Let us take a brief look at their result.

98 Chapter 5. Inductive constructions and decompositions

s π( 1)• • π(s2) T1 • T2

T3 π(s3)• •

Figure 5.1: A matroid-based packing of rooted-arborescences where the set of the inde- S pendent sets of the matroid on S = {s1, s2, s3} is 2 \ S.

Let G = (V,E) be a graph, M a matroid on a ﬁnite set S = {s1,..., st} and π : S → V a map. The quadruple (G, M, S, π) is then called a matroid-based rooted-graph. An S-rooted tree of G is a pair (T, s) where T is a tree of G and s ∈ S with π(s) ∈ V (T ). The element s is then called the root of (T, s) and the set S is called the root set of G. A matroid-based packing of rooted-trees of (G, M, S, π) is a set {(T1, s1),..., (Tt, st)} of pairwise edge-disjoint S-rooted trees of G such that for each v ∈ V , the set {si : v ∈ V (Ti)} forms a base of M. The following undirected counterpart of Theorem 5.4.6 is slightly stronger than that stated in [73] but in fact is implicit in [73].

Theorem 5.4.7 (Katoh and Tanigawa [73]). Let (G, M, S, π) be a matroid-based rooted graph. There exists a matroid-based packing of rooted-trees in (G, M, S, π) if and only if the following conditions are satisﬁed.

1. Sv is independent in M for each v ∈ V , and

2. eG(P) ≥ rM(S)|P| − X∈P rM(SX ), for all partition P of V . P A map π satisfying condition 1 in Theorem 5.4.6 is said to be M-independent.A quadruple (G, M, S, π) satisfying condition 2 in this theorem is said to be partition- connected.

s π( 1)• • π(s2) T1 • T2

T3 π(s3)• •

Figure 5.2: A matroid-based packing of rooted-trees where the set of the independent S sets of the matroid on S = {s1, s2, s3} is 2 \ S.

99 5.4. Packing of matroid-based arborescences

In the same manner as Theorem 5.4.1 is obtained from its directed counter part, Theorem 5.4.2, Theorem 5.4.7 follows from our Theorem 5.4.6 through the following general orientation result of Frank.

V Theorem 5.4.8 (Frank [38]). Let G = (V,E) be a graph and h : 2 → Z+ an intersecting supermodular non-negative non-increasing set-function. There exists an orientation D of G such that ρD(X) ≥ h(X) for all non-empty X ⊂ V if and only if for every partition P of V ,

eG(P) ≥ h(X). X∈P X

Proof of Theorem 5.4.7 from Theorem 5.4.6

First suppose that there exists a matroid-based packing {(T1, s1),..., (Tt, st)} of rooted-trees in (G, M, S, π). Let D be an orientation of G where each rooted- ′ ′ ′ tree (Ti, si) becomes a rooted-arborescence (Ti , si). Then {(T1, s1),..., (Tt , st)} is a matroid-based packing of rooted-arborescences in (D, M, S, π). By Theorem 5.4.6, π is M-independent and (D, M, S, π) is rooted-connected and hence, by Theorem 5.4.8,(G, M, S, π) is partition-connected. Now suppose that π is M-independent and (G, M, S, π) is partition-connected. By Theorem 5.4.8, there exists an orientation D of G such that (D, M, S, π) is rooted-connected. Then, by Theorem 5.4.6, applied to the function h(X) = rM(S) − rM(SX ), there exists a matroid-based packing of rooted-arborescences in (D, M, S, π) which provides, by forgetting the orientation, a matroid-based packing of rooted-trees in (G, M, S, π). 2

In fact, Katoh and Tanigawa deduced Theorem 5.4.7 from its dual form given below. We show that Theorem 5.4.7 also implies Theorem 5.4.9.

Theorem 5.4.9 (Katoh and Tanigawa [73]). Let (G, M, S, π) be a matroid-based rooted-graph. Let M be of rank k with rank function rM. Then (G, M, S, π) admits a matroid-based rooted-tree decomposition if and only if π is M-independent, |E|+

|S| = k|V | and |E(X)| ≤ k|X| − k + rM(SX ) − |SX | for all non-empty X ⊆ V .

Proof. We ﬁrst prove the necessity. The M-independence of π is obviously necessary since each vertex is always covered by the trees rooted at elements s placed at that vertex. Let D be the orientation of G where each rooted-tree in the decomposition is oriented out from its root. Then in each arborescence, each vertex

100 Chapter 5. Inductive constructions and decompositions has in-degree 1 except to the root which has in-degree 0. Moreover, each vertex is covered by exactly k arc-disjoint arborescences, and there are |S| arborescences in total. So the total number of in-degrees of all vertices in D is k|V |−|S| which is also |E|. Therefore |E| + |S| = k|V | holds. Now for each vertex set X ⊆ V , the total number of in-degrees of all vertices in X is v∈X ρD(v) = k|X| − |SX |. Moreover, a vertex v in X can not be covered by more than r (S ) arborescences rooted P M X in vertices of X, thus v must be covered by at least k − rM(SX ) arborescences rooted at vertices in V \ X. Hence, there are at least k − rM(SX ) arcs entering X. Therefore

|E(X)| = ρD(v) − ρD(X) ≤ k|X| − k + rM(SX ) − |SX | v∈X X holds. Now suppose that the conditions hold. For every partition P of V , by the inequality applied for X ∈ P and by |E| + |S| = k|V |, we have

eG(P) = |E| − |E(X)| X X∈P ≥ |E| − (k|X| − k + rM(SX ) − |SX |) X X∈P = k|P| − rM(SX ). X∈P X Hence (G, M, S, π) is partition-connected. Then, since π is M-independent, Theo- rem 5.4.7 implies that (G, M, S, π) admits a matroid-based packing of rooted-trees which, by |E| + |S| = k|V |, must be a matroid-based rooted-tree decomposition of (G, M, S, π).

Contribution and organization: In Section 5.4.2 we provide a proof of our main result in this chapter, Theorem 5.4.6. This proof is short and relatively simple. Therefore, as discussed above, we obtain a short alternative proof for Theorem 5.4.7 and hence Theorem 5.4.9 of Katoh and Tanigawa. In Section 5.4.3 we oﬀer a polyhedral description for the matroid-based packing of rooted-arborescences. Section 5.4.4 considers the algorithmic aspect of our packing problem. We show that the problem of discerning the existence of a matroid-based packing of rooted- arborescences is in P and the corresponding optimization problem also can be solved in polynomial time. Lastly, Section 5.4.5 discusses related problems and a recently extended result based on our result.

101 5.4. Packing of matroid-based arborescences

Results in this section are from a joint work with Olivier Durand de Gevigney and Zolt´an Szigeti [28].

5.4.2 Proof of the main theorem

First we prove the necessity of the conditions.

Proof of necessity in Theorem 5.4.6

Suppose that there exists a matroid-based packing T = {(T1, s1),..., (Tt, st)} of rooted-arborescences in (D, M, S, π). Let v be an arbitrary vertex of V and X a vertex set containing v. Since the root set of all rooted-arborescences covering v is a base of M, the root set of all rooted-arborescences of T covering v with root in X is an independent set, in particular, Sv is independent. Hence, there are at most rM(X) arborescences of T with root in X that cover v. Therefore, there are at least rM(S) − rM(X) rooted-arborescences of T covering v which has root in V \ X. In each of these rooted-arborescences, there is at least one arc entering X. Since these rooted-arborescences are arc-disjoint, the number of arc entering X is at least rM(S) − rM(X), that is (D, M, S, π) is rooted-connected. 2

To prove the suﬃciency, let us introduce the following deﬁnitions. A vertex set X is called tight if ρD(X) = rM(S) − rM(SX ). For vertex sets X and

Y , we say that Y dominates X if SX ⊆ clM(SY ). Note that since, for Q ⊆ S, clM(clM(Q)) = clM(Q), domination is a transitive relation. We say that an arc uv is bad if v dominates u, otherwise it is good. Note that only good arcs are potential candidates for arcs of rooted-arborescences in a matroid-based packing of rooted-arborescences.

Claim 1. Suppose that (D, M, S, π) is rooted-connected. Let X be a tight set and v a vertex of X.

(a) If Y is a tight set that contains v, then X ∩Y and X ∪Y are tight. Moreover,

if s ∈ clM(SX ) ∩ clM(SY ), then s ∈ clM(SX∩Y ).

(b) If no good arc exists in D[X], then v dominates X.

102 Chapter 5. Inductive constructions and decompositions

Proof. (a) We have

ρD(X) + ρD(Y ) = rM(S) − rM(SX ) + rM(S) − rM(SY )

≤ rM(S) − rM(SX ∪ SY ) + rM(S) − rM(SX ∩ SY )

≤ rM(S) − rM(SX∪Y ) + rM(S) − rM(SX∩Y )

≤ ρD(X ∪ Y ) + ρD(X ∩ Y )

≤ ρD(X) + ρD(Y ).

The ﬁrst equility holds by the tightness of X and Y , the second inequality holds by the submodularity of rM, the fourth holds by the rooted connectedness of

(D, M, S, π) and the last holds by the submodularity of ρD. Therefore we have equality everywhere and hence X ∪ Y , X ∩ Y are tight. We also have

rM(SX ) + rM(SY ) = rM(SX ∪ SY ) + rM(SX ∩ SY ) holds.

If s ∈ clM(SX ) ∩ clM(SY ) then

rM(SX∪Y ) + rM(SX∩Y ) = rM(SX ) + rM(SY )

= rM(SX ∪ s) + rM(SY ∪ s)

≥ rM(SX ∪ SY ∪ s) + rM((SX ∩ SY ) ∪ s)

≥ rM(SX∪Y ∪ s) + rM(SX∩Y ∪ s)

≥ rM(SX∪Y ) + rM(SX∩Y ).

Therefore, again, equality holds everywhere, in particular, rM(SX∩Y ∪s) = rM(SX∩Y ) holds, which means that s ∈ clM(SX∩Y ). (b) Let us denote by Y the set of vertices from which v is reachable in D[X]. We show that v dominates Y and Y dominates X and then, since domination is transitive, (b) follows.

For all y ∈ Y , there exists a directed path y = vl, . . . , v1 = v from y to v in

D[X]. Since no good arc exists in D[X], Sy = Svl ⊆ clM(Svl−1 ) ⊆ · · · ⊆ clM(Sv1 ) = clM(Sv). Hence SY = y∈Y Sy ⊆ clM(Sv) and v dominates Y. By the deﬁnition of Y , every arc of D that enters Y enters X as well. Then, by S the rooted-connectedness of (D, M, S, π), the tightness of X and the monotonicity of rM, we have rM(S) − rM(SY ) ≤ ρD(Y ) ≤ ρD(X) = rM(S) − rM(SX ) ≤ rM(S) − rM(SY ). Thus equality holds everywhere and Y dominates X.

Now we can prove the main result.

103 5.4. Packing of matroid-based arborescences

Proof of suﬃciency in Theorem 5.4.6

We prove by induction on the number of good arcs.

First consider the case when no good arc exists. Let Ti be the arborescence containing only on vertex ri = π(si) for i = 1, . . . , t. Then {(Ti, si): i = 1, . . . , t} forms a matroid-based packing of rooted-arborescences in (D, M, S, π). To prove this it is suﬃcient to show that Sv is a base of M for all v ∈ V . Indeed, let Xv be the set of vertices from which v is reachable in D. Then ρD(Xv) = 0 holds. The rooted- connectedness of (D, M, S, π) implies that rM(S) − rM(SX ) = 0. However, since there are only bad arc in (D, M, S, π), rM(SXv ) = rM(Sv). Therefore rM(Sv) = rM(S) holds. Combined with the M-independence of π we obtain that Sv is a base of M.

Now suppose that at least one good arc exists.

For a good arc uv ∈ A and s ∈ Su \cl(Sv) we deﬁne a new matroid-based rooted digraph as follows. We set D′ = D − uv, S′ = S ∪ s′ where s′ is a new element. We extend M to a matroid M′ on S′ by deﬁning s′ as an element parallel to s. Lastly, we obtain a placement π′ of S′ in V from π by placing the new element s′ at v.

••• •••

••• ••• π(s′) v v

• π(s) • π(s) u u

in D in D′

Figure 5.3: Changing rooted-arborescences.

A matroid-based packing of arborescences T ′ of (D′, M′, S′, π′) will provide a matroid-based packing of arborescences T for (D, M, S, π) as follows. Let

(T ′′, s) = (T ∪ T ′ ∪ uv, s), T = T ′ ∪ {(T ′′, s)}) \{(T, s), (T ′, s′)}.

Since s and s′ are parallel in M′, the rooted-arborescences (T, s) and (T ′, s′) of P′ are vertex disjoint, so (T ′′, s) is a rooted-arborescence. Then T ′ is a matroid- based packing of rooted-arborescences in (D, M, S, π).

104 Chapter 5. Inductive constructions and decompositions

Since D′ has less good arcs than D if π′ is M′-independent and (D′, M′, S′, π′) is rooted-connected then (D′, M′, S′, π′) has a matroid-based packing of arborescences by induction. Since the M′-independence of π′ is trivial from the M- independence of π and the fact that s ∈ Su − cl(Sv), for the existence of a matroid- based packing of arborescences in (D′, M′, S′, π′), by induction hypothesis it is ′ ′ ′ ′ suﬃcient to ﬁnd a good arc uv and s ∈ Su \ cl(Sv) such that (D , M , S , π ) is rooted-connected. Assume that such a good arc does not exist. Let uv ∈ A be a good arc ′ ′ ′ ′ and s ∈ Su \ cl(Sv). Then since (D , M , S , π ) is not rooted-connected, there

′ ′ ′ exists ∅= 6 Xs ⊂ V such that ρD (Xs) < rM(S) − rM (SXs ). Hence, by the rooted- connectedness of (D, M, S, π) and the monotonicity of rM′ ,

ρD′ (Xs) + 1 ≥ ρD(Xs)

≥ rM(S) − rM(SXs )

′ ′ ≥ rM(S) − rM (SXs )

≥ ρD′ (Xs) + 1.

So equality holds everywhere and thus uv enters X, Xs is tight in (D, M, S, π) and s ∈ clM(SXs ). Hence, by Claim 1(a), X = ∪s∈Su\cl(Sv )Xs is tight and, by v ∈ X,

Su = (Su \ cl(Sv)) ∪ (Su ∩ cl(Sv)) ⊆ cl(SX ) ∪ cl(SX ) = cl(SX ). So we have proved that

every good arc uv enters a tight set X that dominates u. (5.4.1)

Among all pairs (uv, X) satisfying (5.4.1) choose one with X minimal. Since X dominates u but v does not dominate u, v does not dominate X. Then, by Claim 1(b), there exists a good arc u′v′ in D[X]. Then, by (5.4.1), u′v′ enters a tight set Y that dominates u′. By v′ ∈ X ∩ Y , the tightness of X and Y , u′ ∈ X,

Su′ ⊆ clM(SY ) and Claim 1(a), we have that X ∩ Y is tight and Su′ ⊆ clM(SX∩Y ). Since the good arc u′v′ enters the tight set X ∩ Y that dominates u′ and X ∩ Y is a proper subset of X (since u′ ∈ X \ Y ), this contradicts the minimality of X. 2

5.4.3 Polyhedral description

In this section we provide a polyhedron describing the matroid-based packings of rooted-arborescences. We need the following general result of Frank [37].

105 5.4. Packing of matroid-based arborescences

V Theorem 5.4.10 (Frank [37]). Let D = (V,A) be a digraph, h : 2 → Z+ a non- negative intersecting supermodular set-function such that ρD(X) ≥ h(X) for every X ⊆ V . Then the polyhedron deﬁned by the following linear system is integer:

1 ≥ x(a) ≥ 0 for all a ∈ A, − x(RD(X)) ≥ h(X) for all non-empty X ⊆ V. The following theorem is a corollary of Theorems 5.4.6 and 5.4.10. Theorem 5.4.11. Let (D = (V,A), M, S, π) be a matroid-based rooted-digraph where M is of rank k with rank function rM and suppose that π is M-independent.

The convex hull PM,D of characteristic vectors of the arc sets of the matroid-based packing of rooted-arborescences in (D, M, S, π) is given by the system

1 ≥ x(a) ≥ 0 for all a ∈ A, (5.4.2) − x(RD(X)) ≥ k − rM(SX ) for all non-empty X ⊆ V, (5.4.3) x(A) = k|V | − |S|. (5.4.4)

In particular, there exists a matroid-based packing of rooted-arborescences in (D, M, S, π) if and only if the polyhedron PM,D is not empty. Proof. By Theorem 5.4.10, the polyhedron P determined by the subsystem (5.4.2) and (5.4.3) is integer. For every x in this polyhedron, we have

− x(A) = x(RD(v)) ≥ (k − rM(Sv)) ≥ (k − |Sv|) = k|V | − |S|. (5.4.5) v V v V v V X∈ X∈ X∈ Therefore, (5.4.4) is a valid inequality for this polyhedron P and hence PM,D is a face of P . Thus PM,D is integer. A characteristic vector of a matroid-based packing of rooted-arborescences in (D, M, S, π) obviously belongs to PM,D. Conversely, if ′ x is a vertex of PM,D then it is integer. Let A = {a ∈ A : x(a) = 1} and D′ = (V,A′). Then the matroid-based rooted digraph (D′, M, S, π) is rooted- connected. Together with the assumption that π is M-independent, from Theorem 5.4.6 we deduce that (D′, M, S, π) has a matroid-based packing of arborescences whose set of arcs is A′. Therefore x is the characteristic vector of this packing. The theorem follows.

5.4.4 Algorithmic aspects

In this section we assume that a matroid is given by an oracle for the rank function, that is, when we give the matroid oracle a subset X ∈ S, it will give back the rank of X. The following theorem is a corollary of Theorems 2.4.1 and 5.4.6.

106 Chapter 5. Inductive constructions and decompositions

Theorem 5.4.12. Let (D, M, S, π) be a matroid-based rooted-digraph. A matroid- based packing of rooted-arborescences in (D, M, S, π) or a vertex v certifying that π is not M-independent or a vertex set X certifying that (D, M, S, π) is not rooted- connected can be found in polynomial time.

Proof. By the submodularity of ρD(X) + rM(SX ), Theorem 2.4.1, using the oracle on M and Theorem 5.4.6, we can either ﬁnd a set violating the condition of rooted- connectedness or a vertex certifying that π is not M-independent or certify that there exists a matroid-based packing of rooted-arborescences. In the latter case, a matroid-based packing of rooted-arborescences can be found in polynomial time following the proof of Theorem 5.4.6. Using the oracle, one can test whether an arc is bad or good. When an arc uv is good, for each ′ ′ ′ ′ s ∈ Su \ cl(Sv), determine in polynomial time whether (D , M , S , π ) is rooted-

′ ′ ′ connected using the submodularity of ρD (X) + rM (SX ), the oracle for the rank function rM′ (that is easily computed from rM) and Theorem 2.4.1. The proof of Theorem 2.4.1 shows that either all arcs are bad or we ﬁnd a good arc uv and ′ ′ ′ ′ s ∈ Su \ cl(Sv) such that (D , M , S , π ) is rooted-connected. In the ﬁrst case,

{(v, s): v ∈ V, s ∈ Sv} is the required packing. In the second case, it leads to the computation of a matroid-based packing of rooted-arborescences in (D′, M′, S′, π′) where D′ contains less arcs than D.

− By the submodularity of x(RD(X)) + rM(SX ) and by Theorem 2.4.1, PM,D can be separated in polynomial time. It is a well-known result by Gr¨otschel, Lov´asz and Schrijver [49] that the separation problem and optimization problem are polynomial-time equivalent. Therefore, Theorem 5.4.12 leads to the following consequence.

Theorem 5.4.13. Let (D, M, S, π) be a matroid-based rooted-digraph and c a cost function on the set of arcs of D. If there exists a matroid-based packing of rooted- arborescences in (D, M, S, π) then one of minimum cost can be found in polynomial time.

5.4.5 Further remarks

Theorem 5.4.6 motivates the following extension. Given a matroid-based rooted- digraph (D, M, S, π) where M has rank function rM and a bound b : V → Z, an

(M, b)-packing of rooted-arborescences is a set {(T1, s1),..., (T|S|, s|S|)} of pairwise arc-disjoint rooted-arborescences such that rM({si ∈ S : v ∈ V (Ti)}) ≥ b(v) for

107 5.4. Packing of matroid-based arborescences all v ∈ V . What is the necessary and sufficient condition for the existance of an (M, b)-packing of rooted-arborescences in (D, M, S, π)? When the function b is constant, i.e., b(v) = b for all v ∈ V , by truncating the matroid M to a matroid of rank b, then using Theorem 5.4.6, one can derive a characterization of matroid- based rooted-digraphs admitting an (M, b)-packing of rooted-arborescences. On the other hand, for general b, the problem turns out to be NP-complete since it contains the disjoint Steiner arborescences problem, that is to find 2 arc-disjoint r-arborescences both covering a specified subset of vertices. Basing on the same proof technique, Csaba Király recently extends our result to maximal independent packing of arborescences. For a subset X ⊆ V let P (X) denote the set of vertices u such that there is a directed path from u to a vertex in X, note that P (X) contains X is as a subset. When X = {v} we simply write P (v). A maximal independent packing of arborescences of (D, M, S, π) is a set

{(T1, s1),..., (T|S|, s|S|)} of pairwise arc-disjoint S-rooted arborescences of D such that, for each v ∈ V , the set {si ∈ S : v ∈ V (Ti)} is independent in M and of size rM(SP (v)).

Theorem 5.4.14 (Cs. Kir´aly [74]). A matroid-based rooted-digraph (D, M, S, π) has a maximal independent packing of arborescences if and only if π is M-independent and

ρD(X) ≥ rM(SP (X)) − rM(SX ) holds for each non-empty X ⊆ V .

This result also generalizes the result of Kamiyama, Katoh and Takizawa (The- orem 5.4.4). Moreover, Cs. Kir´aly also pointed out that Theorem 5.4.5 in turn can be obtained easily from Theorem 5.4.4 and therefore these two theorems are infact equivalent.

108 Chapter 6

Inﬁnitesimal rigidity

Contents 6.1 Introduction ...... 110 6.2 Body-bar frameworks with boundary ...... 112 6.2.1 Body-bar frameworks ...... 112 6.2.2 Body-bar frameworks with bar-boundary ...... 115 6.3 Body-length-direction frameworks ...... 119 6.3.1 Introduction ...... 119 6.3.2 Frameworks with length-direction-rigid bodies ...... 120 6.3.3 Frameworks with direction-rigid bodies ...... 122 6.3.4 Frameworks with length-rigid bodies ...... 126 6.4 Direction-length frameworks ...... 131 6.4.1 Introduction ...... 131 6.4.2 0-extensions for direction-length graphs ...... 133 6.4.3 1-extensions for direction-length graphs ...... 135 6.4.4 Union of two spanning trees ...... 141

109 6.1. Introduction

6.1 Introduction

Infinitesimal rigidity plays a central role in rigidity theory. One important reason is that it is more tractable than other rigidity properties. Given a framework, its infinitesimal rigidity can be determined by calculating the rank of a rigidity matrix while determining whether the framework is locally rigid, globally rigid or universally rigid seems difficult [1, 91]. Moreover, infinitesimal rigidity implies local rigidity in general, and when we restrict ourselves to (linearly) generic frameworks, infinitesimal rigidity coincides with local rigidity [9, 65], note that almost all frameworks are (linearly) generic. Infinitesimal rigidity is also highly involved in the study of other rigidity properties. For example, Theorem 3.2.6 states that a stress matrix of maximal rank is a certificate for the global rigidity of an algebraically generic bar-joint framework. But in most cases, it is not easy to find this certificate for a generic framework. A result of Connelly and Whiteley [25] shows that if (G, p) (not necessarily generic) is infinitesimally rigid in Rd and has a stress matrix of maximal rank then G is generically globally rigid in Rd. This property is extremely useful in proving the global rigidity preservingness of certain operations on graphs. Furthermore, infinitesimal rigidity is also important in the neighborhood stability of other rigidity properties such as globally linkedness of pairs of vertices [63]. This chapter contains results on infinitesimal rigidity of frameworks with mixed constraints. In Section 6.2 we begin by reviewing body-bar frameworks, the first model where a complete characterization of infinitesimal rigidity of generic frameworks is obtained. We then consider body-bar frameworks with bar-boundary, i.e, frameworks where some bodies are linked to an external enviroment with bars. We re-obtain a characterization of infinitesimal rigidity of generic frameworks of this type using our decomposition of graded tight graphs in Section 5.2. Section 6.3 introduces the study of body-length-direction frameworks, where several types of bodies that allow different types of motions are considered. In addition, beside distance constraints, these frameworks are also subject to direction constraints. Characterization of infinitesimal rigidity for different types of frameworks are obtained. Section 6.4 considers the so-called direction-length frameworks, an extended model of bar-joint frameworks, where we concern also with direction constraints between vertices. Characterizing infinitesimal rigidity and global rigidity of these frameworks has application in Computer-Aided-Design and network localization.

110 Chapter 6. Inﬁnitesimal rigidity

Complete characterization is obtained by B. Servatius and Whiteley for infinitesimal rigidity in dimension 2 [94] and partial results for global rigidity are obtained by Jackson and Jordán [61]. We introduce extension operations for these frameworks and investigate the infinitesimal rigidity preservingness of these operations in general dimension. Chapter 7 deals with the global rigidity preservingness of these operations.

111 6.2. Body-bar frameworks with boundary

b a

Figure 6.1: A body-bar structure and its corresponding bar-joint graph.

6.2 Body-bar frameworks and body-bar frameworks with bar-boundary

6.2.1 Body-bar frameworks

A body-bar structure is a structure constituting of solid bodies kept together by solid bars. The bodies allow translations and rotations as their motions. A body- bar structure can be modeled by a body-bar framework which is a pair (G, p) where G is a (multi)graph without loops on the body set and p is an embedding of the bars. This framework can be converted to a bar-joint framework by converting each extremity of bars to a joint and adding a complete graph on the set of joints on the same body. Figure 6.1 illustrates a body-bar structure and the corresponding bar- joint graph. However, while the problem of finding a combinatorial characterization for underlying graphs of infinitesimally rigid generic bar-joint frameworks is one central open problem in rigidity theory, the same problem for body-bar frameworks is much easier. Let G = (V,E) be a graph without loops. A realization of G as a body-bar framework is a map p which associates each edge e = uv ∈ E with a pair of points e e d 1 pu, pv ∈ R . The actual shapes and positions of bodies in V are not of significance .

We may think as that each vertex u ∈ V is assigned a d-dimensional body Bu and e pu ∈ Bu for all edges e incident to u. An inﬁnitesimal motion of a body-bar framework (G, p) is a map q that assigns to each body u ∈ V a pair (Au, tu) where Au is a d × d skew-symmetric real matrix d and tu a vector in R such that, for each edge e = uv ∈ E

e e e e h(Aupu + tu) − (Avpv + tv) , pu − pvi = 0

1In fact, in bar-joint frameworks, for the purpose of studying inﬁnitesimal rigidity, only the e e line passing through pu, pv is of signiﬁcance, but not these two points themselves.

112 Chapter 6. Inﬁnitesimal rigidity

(c.f. Section 2.6). d For each u ∈ V , let wu be the 2 -dimensional vector associated with the skew- symmetric matrix Au as deﬁned in Section 2.5. Using equation (2.5.1) we can rewrite this equality as

e e e e e e e e h(pu − pv) ∨ pu , wui − h(pu − pv) ∨ pv , wvi + hpu − pv , tu − tvi = 0.

e e e e e e e e e e Note that (pu − pv) ∨ pu = pu ∨ pv and (pu − pv) ∨ pv = pu ∨ pv, so the equality becomes e e e e hpu ∨ pv , wu − wvi + hpu − pv , tu − tvi = 0. For p, p′ ∈ Rd let T (p, p′) = (p, 1)∨(p′, 1) ∈ RD where (p, 1), (p′, 1) denote the (d+1)- dimensional vectors obtained by adding 1 to p, p′ as the (d + 1)-th coordinates and d+1 D = 2 . Put qu = (wu, tu). The above equality can be rewritten as

e e hT (pu, pv) , qu − qvi = 0.

Therefore, an inﬁnitesimal motion of (G, p) can be regarded as an assignment to D each body u ∈ V of a vector qu ∈ R such that

e e hT (pu, pv) , qu − qvi = 0 for all e = uv ∈ E, u, v ∈ V. (6.2.1)

We can also regard q as a D|V |-dimensional vector obtained by putting qu for u ∈ V consecutively.

An infinitesimal motion of a body-bar framework (G, p) is trivial if qu = qv for all u, v ∈ V . A body-bar framework (G, p) is said to be infinitesimally rigid if all its infinitesimal motions are trivial. The rigidity matrix R(G, p) of a body-bar framework (G, p) is an |E| × D|V | matrix defined as follows. Each edge of E indexes one row of R(G, p) while each vertex of v indexes D columns of R(G, p) in such a way that

• the block indexed by an edge e = uv ∈ E and the vertices u, v are Re,u = e e T (pu, pv) and Re,v = −Re,u, respectively.

• all the other entries are 0’s.

Then equation (6.2.1) implies that Ker R(G, p) is the space of all infinitesimal motions of (G, p). We deduce that (G, p) is infinitesimally rigid if and only if dim Ker R(G, p) is D, which is the dimension of the space of all trivial infinitesimal motions. Equivalently speaking, (G, p) is infinitesimally rigid if and only if the rank of R(G, p) is equal to D|V | − D.

113 6.2. Body-bar frameworks with boundary

A d-dimensional body-bar framework (G, p) is said to be generic if every submatrix of R(G, p) attains the maximum rank among all d-dimensional realizations of G. Then it is not difficult to see that if (G, p) has an infinitesimally rigid realization then all its generic realizations are infinitesimally rigid. Moreover, similarly to the case of bar-joint frameworks, almost all d-dimensional realizations of a graph G as body-bar frameworks are generic. Let e0 = (0,..., 0) ∈ Rd and for i = 1, d let ei denote the ith unit vector of Rd. h k Put Th,k = T (e , e ).

D Lemma 6.2.1. The family of D vectors {Th,k : 0 ≤ h < k ≤ d} is a base of R .

Proof. It is easy to check the following fact on the {i, j} entry of Th,k. For h = 0, 1 ≤ k ≤ d

−1 if i = k, j = d + 1, Th,k{i, j} = ( 0 otherwise.

For 1 ≤ h < k ≤ d,

−1 if i = k, j = d + 1 Th,k{i, j} =  1 if i = h, j = k or i = k, j = d + 1,  0 otherwise.  Suppose that scalars αh,k satisfy

αh,kTh,k = 0, (6.2.2) h

α0,kT0,k = 0. k d 1≤X≤

Recall that T0,k{i, j} 6= 0 if and only if k = i, j = d + 1, by evaluating this equation for entries {i, d+1}, 1 ≤ i ≤ d, we get α0,i = 0 for all 1 ≤ i ≤ d. Therefore, we have shown that αh,k = 0 for all h, k, 0 ≤ h < k ≤ d. That is, {Th,k : 0 ≤ h < k ≤ d} is linearly independent. The lemma follows.2

2 In fact, we can take any p0, . . . , pd aﬃnely independent. It is a well known fact that then D {(ph, 1) ∨ (pk, 1) : 0 ≤ h < k ≤ d} is linearly independent in R .

114 Chapter 6. Inﬁnitesimal rigidity

The following theorem by Tay characterizes graphs that can be realized as inﬁnitesimally rigid body-bar frameworks in Rd.

Theorem 6.2.2 (see [101],[99],[107]). A graph G = (V,E) can be realized as an inﬁnitesimally rigid body-bar framework in dimension d if and only if G contains D edge-disjoint spanning trees.

Proof. We present a proof using ideas from [60][73]. If G can be realized as an inﬁnitesimally rigid d-dimensional body-bar framework (G, p) then rank R(G, p) = D|V | − D. Let H be the spanning subgraph of G whose edge set indexes a maximal linearly independent set of rows in R(G, p). We show that H is (D, D)-tight. Since |E(H)| = D|V | − D, it will suﬃce to show that H is (D, D)-sparse. Let X ⊆ V and F ⊆ E be the edge set of H induced by X. Let M be the submatrix of R(G, p) indexed by F and X. For each i ∈ {1,..., D}, the sum of the ith columns of M indexed by each v ∈ X is 0. Therefore, |F | = rank M ≤ D|X| − D. Now, Theorem 5.1.1 of Nash-Williams implies that H is the edge-disjoint union of D spanning trees.

Conversely, let Fh,k, 0 ≤ h < k ≤ d, be D edge-disjoint spanning trees in G. An infinitesimally rigid d-dimensional realization (G, p) of G can be constructed e h e k as follows. For an edge e = uv ∈ E, if e ∈ Fh,k then let pu = e , pv = e . To an edge e ∈ E that does not belong to any trees Fh,k we can assign an arbitrary pair of points in Rd. We prove that this realization of G is infinitesimally rigid by showing that every infinitesimal motion of (G, p) is trivial. Indeed, let q = (. . . , qu,... ) be an infinitesimal motion of (G, p), then we must have hqu − qv ,Th,ki = 0 for every e = uv ∈ Fh,k, for all 0 ≤ h < k ≤ d. Since Fh,k are spanning trees of G, for every pair of vertices u, v ∈ V , there is a path u = v1, e1, . . . , vs−1, es, vs = v from u to v in Fh,k. On this path we have hqvi − qvi+1 ,Th,ki = 0. Therefore we deduce that

hqu − qv ,Th,ki for all u, v ∈ V and 0 ≤ h < k ≤ d.

D D However, the family of D vectors {Th,k : 0 ≤ h < k ≤ d} ⊆ R is a base of R by

Lemma 6.2.1. Therefore qu − qv = 0 for all u, v ∈ V . The theorem follows.

6.2.2 Body-bar frameworks with bar-boundary

In this model we imagine that some bodies are linked to an external ﬁxed environment by bars. A body-bar framework with bar-boundary is a realization of a graph

115 6.2. Body-bar frameworks with boundary

G which may contain loops where non-loop edges are realized as normal bars in body-bar model and loops are realized as bars that link the incident bodies to the external ﬁxed environment. More precisely, a realization of a graph G = (V,E) as a body-bar framework with bar-boundary is a map p that associates each non-loop e e e e d edge e = uv ∈ E with a join Te = T (pu, pv) where pu, pv ∈ R , and each loop c ∈ E c c c c d with a join Tc = T (p1, p2) of two points p1, p2 in R . An inﬁnitesimal motion of a body-bar framework with bar-boundary (G, p) is D a map q that assigns to each body u ∈ V a vector qu ∈ R such that

• for each non-loop edge e = uv ∈ E,

hTe , qu − qvi = 0,

• for each loop c ∈ E at a vertex u,

hTc , qui = 0.

note that this equation for loops is obtained from the equation for non-loop

edge by setting qv = 0, reﬂecting the fact that the external environment is ﬁxed.

A body-bar framework with bar-boundary (G, p) is said to be infinitesimally rigid if the only infinitesimal motion is qu = 0 for all u ∈ V , namely, it is fixed to the external environment. The rigidity matrix for a body-bar framework with bar-boundary is defined in the same way as that for a body-bar framework, but for a loop c at a vertex u, there is only one non-zero block Rc,u. It is easy to see that a body-bar framework with bar-boundary is infinitesimally rigid if and only if Ker R(G, p) = 0, or equivalently, rank R(G, p) = D|V |. A generic realization of G in Rd as a body-bar framework with bar-boundary is a d-dimensional realization (G, p) such that every submatrix of the rigidity matrix of R(G, p) attains its maximum possible rank for a realization in Rd. It is a common fact that almost all realization of G in Rd are generic. Moreover, if G has an infinitesimally rigid realization in Rd then all its generic realization in Rd are infinitesimally rigid. The following result characterizes graphs that can be realized as infinitesimally rigid d-dimensional body-bar frameworks with bar-boundary.

Theorem 6.2.3. Let G = (V,E) be a graph. Consider the 2-grading E1 = E and

E2 = the set of non-loop edges in E. Then G can be realized as an inﬁnitesimally

116 Chapter 6. Inﬁnitesimal rigidity rigid body-bar with bar-boundary if and only if it contains a (D, (0, D))-graded tight spanning subgraph.

Before proceeding to the proof, let us mention a similar result of Katoh and Tanigawa [73] on body-bar frameworks with pre-determined bar-boundary. In that model, the configuration for loops, i.e., bars that link the framework to the external environment, is given and one wants to know whether there exists a configuration for non-loop edges, i.e., internal bars, such that the framework is infinitesimally rigid. Using their result on packing of matroid-based rooted trees (which can be implied from our result in Section 5.4), Katoh and Tanigawa shows the following result. Let T (Rd) denote the set T (p, p′) for all p, p′ ∈ Rd.

Theorem 6.2.4 ([73]). Let G = (V,F ∪ R) a graph where R is the set of loops of G and F is the set of non-loop edges. Let p◦ : R → T (Rd) a pre-determined configuration for the loops in R. Then there exists a configuration p : E → T (Rd) such that (G, p, p◦) is minimally infinitesimally rigid if and only if

◦ 1. {pr : r ∈ Rv} is linearly independent,

′ ′ ′ ◦ ′ ′ 2. |F | + |RF | ≤ D|V (F )| − D + dim ({pr : r ∈ RF }), for every F ⊆ F ,

3. |F | + |R| = D|V |, where Rv denotes the set of loops at v and RF ′ denotes the set of loops at vertices in V (F ′).

When the pre-determined conﬁguration p◦ is chosen to be generic, our Theorem 6.2.3 coincides with Theorem 6.2.4. It is interesting that two diﬀerent generalizations of Nash-Williams’ theorem give the same result in this case.

Proof of Theorem 6.2.3. Suppose that G = (V,E) has a realization (G, p) in Rd as an infinitesimally rigid body-bar framework with bar-boundary, we show that G has a (D, (0, D))-tight spanning subgraph with the 2-grading defined in the theorem. Since (G, p) is infinitesimally rigid, rank R(G, p) = D|V |. Therefore, there are D|V | linearly independent rows in R(G, p). Let H be a spanning subgraph of G whose edge set indexes a maximal linearly independent set of rows of R(G, p). We show that H is a (D, (0, D))-tight spanning subgraph of G. First, note that |E(H)| = D|X| by the way we choose H. It will suffice to show that H is (D, (0, D))- sparse. Let X ⊆ V and F be the edge set of H induced by X. Let M be the submatrix of R(G, p) indexed by F and X. Then |F | = rank M ≤ D|V |. Now

117 6.2. Body-bar frameworks with boundary

let N be the submatrix of R(G, p) indexed by F ∩ E2 and X. Then for each i ∈ {1,..., D} the sum of the ith column of N indexed by each v ∈ X is 0.

Therefore, |i2(X)| = |F ∩ E2| = rank N ≤ D|X| − D. Conversely, suppose that G contains a (D, (0, D))-tight spanning subgraph H. Then by Theorem 5.2.3, H can be decomposed into D spanning tight pseudoforests

Fh,k, 0 ≤ h < k ≤ d such that in each tight pseudoforest Fh,k there is no cycle containing only non-loop edges. It means that each connected component of Fh,k contains exactly one loop. We construct a d-dimensional realization p of G as follows. For e ∈ E(H), if e ∈ Fh,k for some 0 ≤ h < k ≤ d, then set Te = Th,k. For d e ∈ E(G) \ E(H) we assign to e an arbitrary Te ∈ T (R ). We prove that this realization of G is infinitesimally rigid by showing that every infinitesimal motion of (G, p) is trivial. Let q = (. . . , qu,... ) be an infinitesimal motion of (G, p). In each connected component of Fh,k there is a unique loop at some vertex x. For this loop we have hqx ,Th,ki = 0. On the other hand, using the same argument as that in Theorem 6.2.2 we obtain that, for every u, v in the same connected component of Fh,k as x, the equality hqu − qv ,Th,ki = 0 holds.

Therefore, we deduce that hqu ,Th,ki = 0 for every vertex u since Fh,k is spanning and each of its connected components contains a loop. Hence, we have proved that hqu ,Th,ki = 0 for every vertex u ∈ V and every tight pseudoforest Fh,k. By D D Lemma 6.2.1, the family of D vectors {Th,k, 0 ≤ h < k ≤ d} ⊆ R is a base of R .

Therefore, qu = 0 for every u ∈ V . The theorem follows.

Remark: The above proof of Theorem 6.2.3 illustrates an application of our decomposition of graded sparse graphs. Another way (suggested by Walter Whiteley) to view a body-bar framework with bar-boundary is to consider the external environment as an additional body and apply the body-bar analysis.

118 Chapter 6. Inﬁnitesimal rigidity

6.3 Body-length-direction frameworks

6.3.1 Introduction

Let G = (V ; L, D) be a graph where (L, D) is a bipartition of the edge set of G. We refer to edges in L and D as length edges and direction edges, respectively. Let d ≥ 2 be an integer and ϕ : D → {1, 2, . . . , d − 1}.(G, ϕ) is called a weighted mixed graph.A realization of (G, ϕ) is a map p which associates a pair e e d e e of points pu, pv ∈ R with each e = uv ∈ L, and a triple (pu, pv,Ue) with each e e d d e = uv ∈ D where pu, pv ∈ R and Ue is an ϕ(e)-dimensional subspace of R which e e is orthogonal to pu −pv. We imagine that the realization (G, ϕ, p) is equipped with e a set of d-dimensional bodies Bu, u ∈ V , and that pu ∈ Bu for all edges e incident to u. An infinitesimal motion of (G, ϕ, p) is a map q which assigns an instantaneous e e e e velocity to each body Bu in such a way that (qu − qv) is orthogonal to (pu − pv) e e for each e = uv ∈ L and (qu − qv) is orthogonal to Ue for each e = uv ∈ D, where e e qu is the instantaneous velocity of pu induced by qu. The realization (G, ϕ, p) is infinitesimally rigid if the only infinitesimal motions of (G, ϕ, p) are translations. We will consider three different types of bodies which allow three different kinds of instantaneous velocities for each body.

1. Length-direction-rigid bodies which keep the distances and directions between all pairs of points fixed. These allow only translations as instantaneous velocities. 2. Direction-rigid bodies which keep the directions between all pairs of points fixed. These allow only translations and dilations as instantaneous velocities. 3. Length-rigid bodies which keep the distances between all pairs of points fixed. These allow only translations and rotations as instantaneous velocities.

Let G = (V ; L, D) and ϕ : D → {1, 2, . . . , d − 1}. We consider the graph Gϕ = (V,L ∪ Dϕ) obtained from G by replacing each direction edge e ∈ D with ϕ(e) copies of itself. We call this graph the augmented graph of G by ϕ. We will also refer to edges in L and Dϕ of Gϕ as length edges and direction edges, respectively.

Contribution: We give a complete characterization of inﬁnitesimal rigidity of generic body-length-direction frameworks with length-direction-rigid bodies and

119 6.3. Body-length-direction frameworks those with direction-rigid bodies (Theorems 6.3.2 and 6.3.3). We obtain a necessary condition for the inﬁnitesimal rigidity of generic body-length-direction frameworks and show that it is also suﬃcient in the special case where ϕ(e) = 1 for all e ∈ D.

Results in this section are from a joint work with Bill Jackson [66].

We will need the following elementary result concerning the rank of a modi- ﬁed edge/vertex incidence matrix for a pseudoforest. We consider the entries as elements of F2, the ﬁeld of integers modulo two, and denote this rank by rank 2.

Lemma 6.3.1. Let P = (V,E) be a pseudoforest and F ⊆ E such that each cycle in P contains an edge of F . Choose an orientation for P such that each cycle is a directed cycle. Let M be the |E| × |V | matrix with rows indexed by E and columns indexed by V , in which:

• the entries in the row indexed by an edge e = uv oriented from u to v, and the vertices u and v are 1, 0 if e ∈ F , and 1, 1 if e∈ / F ;

• all other entries are zero.

Then rank 2M = |E|.

Proof. If P is not connected then we may apply the lemma inductively to each connected component of P . Similarly, if P has a vertex of v of degree one, then we may apply the lemma inductively to P −v. Hence we may assume that P is a cycle, say P = v1e1v2e2 . . . en−1vnenv1, and we may suppose further that P is oriented in this cyclic order. Since P has at least one edge in F , we may also suppose that en ∈ F . This makes M an upper triangular matrix with ones down the diagonal and the lemma follows.

6.3.2 Frameworks with length-direction-rigid bodies

We consider realizations of (G, ϕ) as a d-dimensional body-length-direction framework with length-direction-rigid bodies. We may assume that G has no loops. Let (G, ϕ, p) be such a realization. An inﬁnitesimal motion q of (G, ϕ, p) can be de- d scribed by assigning to each body u ∈ V , a vector qu ∈ R such that (qu − qv) is e e orthogonal to (pu − pv) for each e = uv ∈ L and (qu − qv) is orthogonal to Ue for each e = uv ∈ D.

120 Chapter 6. Inﬁnitesimal rigidity

It is convenient to think of p as assigning a vector mf ∈ Rd to each edge f of Gϕ such that for each e ∈ D, the set of all vectors mei assigned to the ϕ(e) copies e e d of e are linearly independent. We can then choose pu, pv ∈ R for each e ∈ D ∪ L e e e e e such that m = pu − pv for each e = uv ∈ L, and such that pu − pv is orthogonal to ei Ue = hm : 1 ≤ i ≤ ϕ(e)i for each e = uv ∈ D, without aﬀecting the inﬁnitesimal rigidity of (G, ϕ, p). The rigidity matrix R(G, ϕ, p) of (G, ϕ, p) is the (|L| + |Dϕ|) × d|V | matrix where each edge of Gϕ indexes one row and each vertex indexes d columns so that

• the submatrices indexed by an edge f = uv and the vertices u and v are mf and −mf , respectively;

• all other entries are 0’s.

Then q is an infinitesimal motion of (G, ϕ, p) if and only if q ∈ Ker R(G, ϕ, p). Thus (G, ϕ, p) is infinitesimally rigid if and only if all infinitesimal motions q satisfy qu = qv for all u, v ∈ V or, equivalently, rank R(G, ϕ, p) = d|V | − d.

Theorem 6.3.2. A weighted mixed graph (G, ϕ) has a realization as an inﬁnites- imally rigid d-dimensional body-length-direction framework with length-direction- rigid bodies if and only if Gϕ has a (d, d)-tight spanning subgraph.

Proof. Suppose that (G, ϕ) has an infinitesimally rigid realization (G, ϕ, p) in Rd. Then rank R(G, ϕ, p) = d|V | − d. Hence R(G, ϕ, p) contains d|V | − d independent rows. Let H be the spanning subgraph of Gϕ whose edge set indexes these rows. We show that H is (d, d)-tight. Since |E(H)| = rank R(G, ϕ, p) = d|V | − d, it is sufficient to show that H is (d, d)-sparse. Let X ⊆ V and let F be the set of edges of H induced by X. Let M be the submatrix of R(G, ϕ, p) indexed by F and X. For each i ∈ {1, 2, . . . , d}, the definition of R(G, ϕ, p) implies that the sum of the ith columns of M indexed by each v ∈ X is 0. Hence |F | = rank M ≤ d|X| − d. This implies that H is (d, d)-sparse. For sufficiency, we will use the decomposition of a (d, d)-tight spanning subgraph of Gϕ into spanning trees to construct an infinitesimally rigid realization for (G, ϕ). Let H be a (d, d)-tight spanning subgraph of Gϕ. By Theorem 5.1.1, H is the edge- 1 2 d disjoint union of d spanning trees T1,...,Td. Let e , e ,..., e denote the canonical base of Rd, i.e., ei is the d-dimensional vector whose ith entry is 1 and all other f i entries are 0’s. For i = 1, . . . , d, and edges f ∈ E(Ti), we set m = e . To each f d f∈ / E(T1) ∪ · · · ∪ E(Td) we assign an m ∈ R such that, for each e ∈ D, the set ei ϕ {m : ei is a copy of e in G } is linearly independent. Recall that each edge of

121 6.3. Body-length-direction frameworks

Gϕ indexes one row of of R(G, ϕ, p) and each vertex of Gϕ indexes d columns of R(G, ϕ, p). If we group together the rows corresponding to the edges in each tree

Ti and the jth columns corresponding to each vertex v ∈ V then R(G, ϕ, p) takes the form V 1 V 2 V d

T1 A1 T2  A2 O  . . . O ..     Td Ad     ∗ ∗ ∗ ∗      where O’s are zero matrices and Ai is the incident matrix of the spanning tree Ti, i = 1, . . . , d. The rank of each Ai is then |V | − 1 and hence

rank R(G, ϕ, p) ≥ rank Ai = d|V | − d. i=1 X Therefore (G, ϕ, p) is inﬁnitesimally rigid. The theorem now follows.

6.3.3 Frameworks with direction-rigid bodies

We consider realizations of (G, ϕ) as frameworks with direction-rigid bodies. We can assume that no direction edge in G is a loop (but length edge loops may exist). An inﬁnitesimal motion of a d-dimensional realization (G, ϕ, p) can be described by assigning, to each body u ∈ V , a pair (tu, ru) where tu is a vector in d R (which corresponds to a translation) and ru is a scalar (which corresponds to a dilation) such that:

• for each non-loop length edge e = uv ∈ L,

e e e e h(tu + rupu) − (tv + rvpv) , pu − pvi = 0, (6.3.1)

• for each length edge loop e = uu ∈ L,

ru = 0, (6.3.2)

• for each direction edge e = uv ∈ D,

e e h(tu + rupu) − (tv + rvpv) , mi = 0, (6.3.3)

for every vector m ∈ Ue.

122 Chapter 6. Inﬁnitesimal rigidity

Equation (6.3.1) can be rewritten as

e e e e e e e e htu − tv , pu − pvi + ruhpu , pu − pvi + rvhpv , pv − pui = 0. (6.3.4)

e e e e e Let p = (pu + pv)/2. Since pu − pv is orthogonal to Ue, equation (6.3.3) can be rewritten as e e htu − tv , mi + ruhp , mi − rvhp , mi = 0 (6.3.5) for all m ∈ Ue. Therefore, for the purpose of studying inﬁnitesimal rigidity, a realization (G, ϕ, p) can be described by assigning, to each non-loop edge e = uv ∈ L, a triple e e e d d d e e e (pu, pv, m ) ∈ R × R × R such that m is multiple of pu − pv, and, to each edge e d e e e = uv ∈ D, a point p ∈ R and ϕ(e) linearly independent vectors m1, . . . , mϕ(e) ∈ Rd. The rigidity matrix R(G, ϕ, p) of (G, ϕ, p) is the (|L|+|Dϕ|)×(d+1)|V | matrix where each length edge indexes one row, each direction edge indexes ϕ(e) rows and each vertex indexes d + 1 columns so that:

• the matrices indexed by a non-loop edge e = uv ∈ L and the vertices u and e e e e e e v are Re,u = (m , hpu , m i) and Re,v = (−m , −hpv , m i), respectively;

• the matrix indexed by a loop edge e ∈ L at a vertex v is Re,v = (0,..., 0, 1);

• the matrices indexed by an edge e = uv ∈ D and the vertices u and v are

Re,u and Re,v = −Re,u, respectively, where

e e e m1 hp , m1i . . Re,u =  . .  ; e e e mϕ(e) hp , mϕ(e)i     • elsewhere all the entries are 0’s.

Then Ker R(G, ϕ, p) is the space of all the inﬁnitesimal motions of (G, ϕ, p) and therefore (G, ϕ, p) is rigid if and only if rank R(G, ϕ, p) = (d + 1)|V | − d.

Theorem 6.3.3. Let G = (V ; L, D) be a mixed graph and ϕ : D → {1, . . . , d − 1}. ϕ Deﬁne a 2-grading (E1,E2) for the augmented graph G by letting E1 be the set of ϕ ϕ all edges of G and E2 be the set of all direction edges in G . Then (G, ϕ) has a d- dimensional realization as an inﬁnitesimally rigid body-length-direction framework with direction-rigid bodies if and only if Gϕ has a (d + 1; (d, d + 1))-graded-tight spanning subgraph.

123 6.3. Body-length-direction frameworks

Proof. First suppose that (G, ϕ, p) is an infinitesimally rigid realization of (G, ϕ) in Rd. Let H be a spanning subgraph of Gϕ whose edge set indexes a maximal linearly independent set of rows in the rigidity matrix R(G, ϕ, p). We show that H is a (d + 1; (d, d + 1))-graded-tight spanning subgraph of Gϕ. Since |E(H)| = rank R(G, ϕ, p) = (d + 1)|V | −d, it is sufficient to show that H is (d + 1; (d, d + 1))- graded sparse. Let X ⊂ V and let F be the set of edges of H induced by X. Let M be the submatrix of R(G, ϕ, p) indexed by F and X. For each i ∈ {1, 2, . . . , d}, the definition of R(G, ϕ, p) implies that the sum of the ith columns of M indexed by each v ∈ X is 0. Hence |F | = rank M ≤ d|X| − d. Similarly, if N is the submatrix of R(G, ϕ, p) indexed by F ∩ E2 and X, then the sum of the ith columns of N indexed by each v ∈ X is 0 for all i ∈ {1, 2, . . . , d + 1}. Hence |F ∩ E2| = rank N ≤ d|X| − d − 1. This implies that H is (d + 1; (d, d + 1))-graded sparse. For sufficiency, we will use the decomposition of a (d+1; (d, d+1))-graded-tight spanning subgraph of Gϕ into spanning pseudoforests to construct an infinitesimally rigid realization R(G, ϕ, p) for (G, ϕ). Let H be a (d + 1; (d, d + 1))-graded- tight spanning subgraph of Gϕ. By Theorem 5.2.3, H is the edge-disjoint union of d 1 2 d spanning trees T1,...,Td and one spanning tight pseudoforest P . Let e , e ,..., e denote the canonical base of Rd, and 1 = e1 + ... + ed be the all one vector in Rd. f i f For f ∈ E(H), we set m = e if f ∈ E(Ti) for some 1 ≤ i ≤ d and m = 1 if f ∈ E(P ). To each f∈ / E(H) we assign an mf ∈ Rd such that, for each e ∈ D, the ei ϕ set {m : ei is a copy of e in G } is linearly independent. We associate each row of R(G, ϕ, p) with an edge of Gϕ in the obvious way. If we group together the rows corresponding to the edges in each pseudoforest and the jth columns corresponding to each vertex v ∈ V then R(G, ϕ, p) takes the form

V 1 V 2 V 3 V d V d+1 ′ T1 A1 OO...OA1 ′ T2  OA2 O...OA2  ′ T3 OOA ...OA  3 3  .  . . . . .  .  . . . . .   ′  Td OOO...A A   d d   ′  P  BBB...BB     ∗ ∗ ∗ ∗ ∗ ∗      where O’s are zero matrices, Ai is the edge/vertex incident matrix of Ti for i = ′ ′ 1, . . . , d, B is the edge/vertex incident matrix of P , and Ai and B are ‘modiﬁed’ edge/vertex incidence matrices for Ti and P , respectively. (Their non-zero entries

124 Chapter 6. Inﬁnitesimal rigidity occur only in positions which contain non-zero entries in the corresponding inci- e e e dence matrices but the values depend on pu, pv for e = uv ∈ L and p for e ∈ D, which we will choose next.) ϕ For each e ∈ D with copies e1, e2, . . . , eϕ(e) in G , choose an index ke ∈ e ke {1, 2, . . . d − 1} such that ej 6∈ Tke for all 1 ≤ j ≤ ϕ(e) and put p = e . For e i e ′ e = uv ∈ L ∩ Ti let pu = 2e and pv = 0. The resulting entries in Ai are as follows.

• The entries indexed by a direction edge ej = uv in Ti, which is a copy of an edge e ∈ D, and the vertices u and v are hpe , mei = heke , eii = 0 and −hpe , mei = 0, respectively.

• The entries indexed by a length edge e = uv in Ti and the vertices u and v e e i i e e are hpu , m i = h2e , e i = 2 and −hpv , m i = −h0 , 1i = 0, respectively.

• Elsewhere all the entries are 0’s.

e e Finally, we specify pu, pv for e ∈ L ∩ P . We ﬁrst choose an orientation of the tight pseudoforest P such that each cycle in P is a directed cycle. For each non-loop e 1 length edge e = uv of P which has been oriented from u to v, put pu = d 1 and e ′ pv = 0. The resulting entries in B are as follows. • The entries indexed by a non-loop length edge e = uv in P which has been e e 1 oriented from u to v, and the vertices u and v are hpu , m i = h d 1 , 1i = 1 e e and −hpv , m i = −h0 , 1i = 0, respectively.

• The entry indexed by a loop length edge e = vv in P and the vertex v is 1.

• The entries indexed by a direction edge e1 in P which is a copy of an edge e e ke e = uv ∈ D, and the vertices u and v are hp , m1i = he , 1i = 1 and e e −hp , m1i = −1, respectively.

• Elsewhere all the entries are 0’s.

′ In particular all the entries of R(G, ϕ, p) are integers and all the entries of Ai, ′ 1 ≤ i ≤ d, are even integers. Since Ai = O over F2, we can now use Lemma 6.3.1 to deduce that

d ′ rank R(G, ϕ, p) ≥ rank 2R(G, ϕ, p) ≥ rank 2Ai + rank 2B i=1 X = d|V | − d + |V | = (d + 1)|V | − d.

Hence (G, ϕ, p) is inﬁnitesimally rigid.

125 6.3. Body-length-direction frameworks

Note that a direction-rigid body becomes length-direction-rigid if we add any length constraint to it. This simple observation allows us to determine whether a generic body-length-direction framework with a mixture of direction-rigid and direction-length-rigid bodies is inﬁnitesimally rigid. We just apply Theorem 6.3.3 to the mixed graph we obtain by adding a length edge loop at the vertices which correspond to direction-length-rigid bodies. In particular, we can use this approach to deduce Theorem 6.3.2 from Theorem 6.3.3.

6.3.4 Frameworks with length-rigid bodies

We consider realizations of (G, ϕ) as frameworks with length-rigid bodies. We may assume that no length edge is a loop in G (but direction edge loops may exist). An inﬁnitesimal motion of such a realization (G, ϕ, p) is an assignment, to each body d u ∈ V , of a pair (tu,Au) where tu is a vector in R (corresponding to a translation) and Au is a d × d skew symmetric matrix (corresponding to a rotation) such that:

• for a length edge e = uv,

e e e e htu + Aupu − (tv + Avpv) , pu − pvi = 0; (6.3.6)

• for a non-loop direction edge e = uv,

e e htu + Aupu − (tv + Avpv) , mi = 0, (6.3.7)

for every vector m in Ue.

• for a loop direction edge e = uu,

e e hAu(pu,1 − pu,2) , mi = 0, (6.3.8)

for all m ∈ Ue.

(See Section 2.6.) d For each u ∈ V , let wu be the 2 -dimensional vector associated with the skew- symmetric matrix Au deﬁned as in Section 2.5. Equation (6.3.6) can be rewritten as

e e e e e e e e htu − tv , pu − pvi + hAupu , pu − pvi − hAvpv , pu − pvi = 0, which is then equivalent to

e e e e e e e e htu − tv , pu − pvi + hwu , (pu − pv) ∨ pui − hwv , (pu − pv) ∨ pvi = 0. (6.3.9)

126 Chapter 6. Inﬁnitesimal rigidity

e e e e e e If we put m = pu − pv and p = (pu + pv)/2 then equation (6.3.9) is equivalent to

e e e e e htu − tv , m i + hwu , m ∨ p i − hwv , m ∨ p i = 0. (6.3.10)

Similarly, equations (6.3.7) and (6.3.8) can be rewritten as

e e htu − tv , mi + hwu , m ∨ pui − hwv , m ∨ pvi = 0 (6.3.11) and e e hwu , m ∨ (pu,1 − pu,2)i = 0 (6.3.12) for each m ∈ Ue. It follows that, for the purpose of studying inﬁnitesimal rigidity, we can think of p as assigning, to each length edge e ∈ L, a pair (pe, me) ∈ Rd × Rd, and, to e e d d each direction edge e = uv ∈ D, a pair (pu, pv) ∈ R × R and ϕ(e) independent d e e vectors m1, . . . , mϕ(e) ∈ R orthogonal to pu − pv. The rigidity matrix R(G, ϕ, p) ϕ d+1 of this realization is a (|L| + |D |) × 2 |V | matrix where each length edge e ∈ L indexes one row, each direction edge e ∈ D indexes ϕ(e) rows and each vertex d+1 indexes 2 columns such that: • the submatrices indexed by a length edge e = uv ∈ L and the vertices u and v are Re = (me me ∨ pe) and −Re, respectively;

• the submatrices indexed by a non-loop direction edge e = uv ∈ D and the vertices u and v are

e e m1 m1 ∨ pu −m1 −m1 ∨ pv e . . e . . Ru =  . .  and Rv =  . .  m m ∨ pe −m −m ∨ pe  ϕ(e) ϕ(e) u  ϕ(e) ϕ(e) v     respectively;

• the submatrix indexed by a loop direction edge e = uu ∈ D and the vertex u is e e 0 m1 ∨ (pu,1 − pu,2) e . . Ru =  . .  0 m ∨ (pe − pe )  ϕ(e) u,1 u,2    • elsewhere all entries are 0’s.

Then Ker R(G, ϕ, p) is the space of all the inﬁnitesimal motions of (G, ϕ, p) and therefore (G, ϕ, p) is inﬁnitesimally rigid if and only if rank R(G, ϕ, p) =

127 6.3. Body-length-direction frameworks

d+1 2 |V | − d. The following result gives a necessary condition for a graph (G, ϕ) to have an inﬁnitesimally rigid realization as a d-dimensional framework with length-rigid bodies.

Theorem 6.3.4. Let G = (V ; L, D) be a mixed graph and ϕ : D → {1, . . . , d − 1}. ϕ Deﬁne a 2-grading (E1,E2) for the augmented graph G by letting E1 be the set of ϕ ϕ all edges of G and E2 be the set of all length edges in G .

(a) If (G, ϕ) has a d-dimensional realization as an inﬁnitesimally rigid body- length-direction framework with length-rigid bodies then the augmented graph ϕ d+1 d+1 G has a ( 2 , (d, 2 ))-graded-tight spanning subgraph.

ϕ d+1 d+1 (b) If ϕ(e) = 1 for all e ∈ D and G has a ( 2 , (d, 2 ))-graded-tight spanning subgraph then (G, ϕ) has a d-dimensional realization as an inﬁnitesi- mally rigid body-length-direction framework with length-rigid bodies.

Proof. (a) Suppose that (G, ϕ, p) is an infinitesimally rigid realization of (G, ϕ) in Rd. Let H be a spanning subgraph of Gϕ whose edge set indexes a maximal linearly independent set of rows in the rigidity matrix R(G, ϕ, p). We show that d+1 d+1 ϕ H is a ( 2 , (d, 2 ))-graded-tight spanning subgraph of G . Since |E(H)| = rank R(G, ϕ, p) = d+1 |V | − d, it is sufficient to show that H is ( d+1 , (d, d+1 ))- 2 2 2 graded sparse. LetX ⊂ V and let F be the set of edges of H induced by X. Let M be the submatrix of R(G, ϕ, p) indexed by F and X. For each i ∈ {1, 2, . . . , d}, the definition of R(G, ϕ, p) implies that the sum of the ith columns of M indexed d+1 by each v ∈ X is 0. Hence |F | = rank M ≤ 2 |X| − d. Similarly, if N be the submatrix of R(G, ϕ, p) indexed by F ∩ E2and X, then the sum of the ith d+1 columns of N indexed by each v ∈ X is 0 for all i ∈ {1, 2,..., 2 }. Hence |F ∩ E | = rank N ≤ d+1 |X| − d+1 . This implies that H is ( d+1 , (d, d+1 ))- 2 2 2 2 2 graded sparse. ϕ d+1 d+1 (b) Suppose that ϕ(e) = 1 for all e ∈ D and G has a ( 2 , (d, 2 ))-graded- ϕ tight spanning subgraph H. Then G = G and H is a spanning subgraph of G. Hence it will suffice to show that H has a d-dimensional realization as an infinitesimally rigid body-length-direction framework with length-rigid bodies. By d Theorem 5.2.3, H is the edge-disjoint union of d spanning trees T1,...,Td and 2 1 2 d spanning tight pseudoforests Pi,j, 1 ≤ i < j ≤ d. Let e , e ,..., e denote the d canonical base of R . Choose an orientation for each tight pseudoforest Pi,j such

128 Chapter 6. Inﬁnitesimal rigidity

that each cycle in Pi,j is a directed cycle. We next choose a realization p for H as follows.

f i • For f = uv ∈ E(Ti), we set m = e . We also choose jf ∈ {1, 2, . . . , d}\{i}

f jf f jf f and put p = 2e if f ∈ L, and pu = 2e and pv = 0 if f ∈ D.

f i j f 1 i j • For f = uv ∈ E(Pi,j), we set m = e − e . We also put p = 2 (e + e ) if f 1 i j f f ∈ L, and pu = 2 (e + e ) and pv = 0 if f ∈ D and f is oriented from u to f f f f v in Pi,j (taking pu, pv to be pu,1, pu,2 respectively, when f = uu is a loop). We index the columns of R(H, p) corresponding to each vertex v ∈ V by vi, 1 ≤ i ≤ d and vi,j, 1 ≤ i < j ≤ d. If we group together the rows corresponding to the edges in each pseudoforest and the columns indexed by the same copy of each v ∈ V then R(H, p) takes the form

V 1 V 2 V 3 V d ′ T1 A1 OO...OA1 ′ T2  OA2 O...OA2  ......    ′  Td  OOO...Ad Ad    P1,1  ∗ ∗ ∗ ... ∗ B1,1    .  . . . . .  .  . . . . .    Pd−1,d  ∗ ∗ ∗ ... ∗ B   d−1,d    where O’s are zero matrices and Ai is the edge/vertex incident matrix of Ti for i = 1, . . . , d. Furthermore, the entries in R(H, p) are all integers, the entries in ′ ′ A1,...,Ad are all even integers, and

′ B1,1 B1,1 O...O ′  B1,2   OB1,1 ...O  = . . . .  .   . . .     ′   Bd−1,d   OO...B     d−1,d  ′     ′ where Bi,j is a modiﬁed incidence matrix for Pi,j. The entries of Bi,j in the row indexed by a non-loop edge f = uv of Pi,j and the columns indexed by the vertices u, v are: 1, −1 if f ∈ L; 1, 0 if f ∈ D and is oriented from u to v. The entry in the row indexed by a loop direction edge f = uu and the column indexed by u is 1. All other entries are zero. We can now use Lemma 6.3.1 to deduce as in the proof of Theorem 6.3.3 that d d + 1 rank R(H, p) ≥ rank R(H, p) ≥ d(|V | − 1) + |V | = |V | − d 2 2 2 129 6.3. Body-length-direction frameworks and hence (H, p) is inﬁnitesimally rigid.

We conjecture that the hypothesis “ϕ(e) = 1 for all e ∈ D” is not needed in ϕ d+1 d+1 (b), and hence that the condition “G has a ( 2 , (d, 2 ))-graded-tight spanning subgraph” is both necessary and suﬃcient for (G, ϕ) to have a d-dimensional inﬁnitesimally rigid realization for all ϕ.

130 Chapter 6. Inﬁnitesimal rigidity

6.4 Direction-length frameworks

6.4.1 Introduction

A direction-length graph is a multigraph G = (V,D ∪L) without loops in which the edge set is partitioned into two sets D and L such that neither D nor L contains parallel edges. Edges in D are called direction edges and edges in L are called length edges.A d-dimensional direction-length framework is a pair (G, p) of a direction- length graph G and a map p which maps each v ∈ V to a point p(v) in Rd. We can also regard p as a (row) vector in Rnd, where n = |V | is the number of vertex of G. We say that (G, p) is a direction-length realization of G in Rd. To illustrate direction-length frameworks we will use dashed segments to denote direction edges and continuous segments to denote length edges. Two realizations (G, p) and (G, q) of G in Rd are equivalent if kp(u) − p(v)k = kq(u) − q(v)k for every length edge uv and there exists a scalar λuv such that p(u) − p(v) = λuv(q(u) − q(v)) for every direction edge uv. They are congruent if there exists a λ ∈ {1, −1} such that p(u) − p(v) = λ(q(u) − q(v)) for every u, v in V , i.e. (G, q) can be obtained from (G, p) by a translation and, possibly, a dilation by −1. A direction-length framework (G, p) is locally rigid if there exists an ǫ > 0 such that, for every equivalent realization (G, q) of (G, p), if kp − qk < ǫ then (G, q) is congruent to (G, p). Equivalently, every continuous motion of (G, p) respecting direction and length constraints results in a framework that is congruent to (G, p). An inﬁnitesimal motion of a direction-length framework (G, p) is an assignment to each vertex v ∈ V a vector µ(v) ∈ Rd such that

(µ(u) − µ(v))(p(u) − p(v))= 0 for each length edge uv ∈ L, µ(u) − µ(v) is parallel to p(u) − p(v) for each direction edge uv ∈ D.

For ease of notation, let [p(u) − p(v)]⊥ denote a (d − 1) × d matrix whose rows form a base of the subspace of Rd orthogonal to p(u) − p(v). We may suppose throughout the sequel that the rows of this matrix are orthonormal. The rigidity matrix R(G, p) is deﬁned as follows. R(G, p) is a (|L| + (d − 1)|D|) × d|V (G)| matrix. Each vertex of G indexes d columns of R(G, p), each length edge xy ∈ L indexes one row in R(G, p) and each direction edge uv ∈ D indexes (d − 1) rows of R(G, p) in such a way that,

• the submatrix of R(G, p) indexed by an edge xy ∈ L and the vertices x, y

131 6.4. Direction-length frameworks

are p(x) − p(y) and p(y) − p(x) respectively; • the submatrix of R(G, p) indexed by an edge uv ∈ D and the vertices u, v are [p(u) − p(v)]⊥ and −[p(u) − p(v)]⊥ respectively; • elsewhere all entries are 0’s.

Here p(x) − p(y), p(u) − p(v) are regarded as row vectors. That is

x y u v ...... xy∈L  p(x) − p(y) p(y) − p(x) 0 0  R(G, p) = ⊥ ⊥ . uv∈D 0 0 [p(u) − p(v)] −[p(u) − p(v)]    . . . .   . . . .      Then it is easy to see that µ is an inﬁnitesimal motion of (G, p) if and only if R(G, p)µ = 0, where µ is regarded as an nd-dimensional column vector. A realization (G, p) in Rd is said to be (linearly) generic if the rank of every submatrix of R(G, p) is the maximum among all realizations of G in Rd. The relation between the rank of the rigidity matrix and the local rigidity of a generic framework is revealed in the following theorem.

Theorem 6.4.1 (Jackson and Keevash [64]). Let G be a direction-length graph.

1. For every embedding q of G into Rd, rank R(G, q) ≤ d|V (G)| − d. More- over, if rank R(G, q) = d|V (G)| − d then for every generic embedding p, rank R(G, p) = d|V (G)| − d. 2. For every generic embedding p, (G, p) is locally rigid if and only if rank R(G, p) = d|V (G)| − d.

This theorem implies that local rigidity is a generic property of direction-length frameworks. We say that a direction-length graph G is generically locally rigid in Rd if some (and therefore every) linearly generic realization of G in Rd as direction- length framework is locally rigid. Servatius and Whiteley [94] characterized generically locally rigid direction- length graph in R2 in terms of sparsity.

Theorem 6.4.2 (Servatius and Whiteley [94]). A direction-length graph G = (V,D ∪ L) is generically locally rigid in R2 if and only if there is a subset F of edges of G satisfying the following conditions.

132 Chapter 6. Inﬁnitesimal rigidity

1. |F | = 2|V | − 2.

2. For every F ′ ⊆ F , |F ′| ≤ 2|V (F ′)| − 2.

3. For every F ′ ⊆ F such that F ′ contains only length edges or only direction edges, |F ′| ≤ 2|V (F ′)| − 3. The main ingredient in the proof of this characterization is an inductive construction of the direction-length graph (V,F ) by 0-extensions and 1-extensions. The 2-dimensional 0-extensions and 1-extensions for direction-length graphs are deﬁned in exactly the same way as those for bar-joint graphs (i.e., length graphs), keeping in mind that the new edges may be direction edges or length edges.

Contribution: We extend the definition of 0-extensions and 1-extensions to d- dimensional case. We show a condition under which these operations preserve the generic local rigidity of direction-length graphs in Rd. Although these operations are not sufficient to characterize generically locally rigid direction-length graphs in general dimension, we obtain a special sufficient condition for a direction-length graph to be generically locally rigid in every dimension, namely, when the direction- length graph contains both a direction spanning tree and a length spanning tree.

6.4.2 0-extensions for direction-length graphs

Deﬁnition 6.4.1. Let H be a direction-length graph. A length-pure d-dimensional 0-extension of H is a direction-length graph G obtained from H by adding a new vertex v and connecting v to d distinct vertices v1, . . . , vd of H by length edges (Figure 6.2). A d-dimensional direction-pure 0-extension of H is a direction-length graph G obtained from H by adding a new vertex v and connecting v to 2 distinct vertices v1, v2 of H by direction edges (Figure 6.3). A d-dimensional mixed 0-extension of H is a direction-length graph G obtained from H by adding a new vertex v and connecting v to 2 vertices v1, v2 of H by a length edge and a direction edge (Figure 6.4). When there is no need to specify, we use “0-extension” to refer to any kind of 0-extensions deﬁned above. The following result shows that d-dimensional 0- extensions preserve generic local rigidity of direction-length graphs. Proposition 6.4.3. Let H be a generically locally rigid direction-length graph in Rd. Suppose that G is a d-dimensional 0-extension of H. Then G is generically locally rigid in Rd.

133 6.4. Direction-length frameworks

v •

length-pure • • ... • • • ... • v1 v2 vd 0-extension v1 v2 vd

Figure 6.2: Length-pure d-dimensional 0-extension. v •

direction-pure • • • • v1 v2 0-extension v1 v2

Figure 6.3: Direction-pure d-dimensional 0-extension. v •

mixed • • • • v1 v2 0-extension v1 v2

Figure 6.4: Mixed d-dimensional 0-extension.

Proof. Let p be a generic embedding of G in Rd. By Theorem 6.4.1, it suﬃces to show that rank R(G, p) ≥ d|V (G)| − d. We consider each possibility. Case 1: G is a length-pure d-dimensional 0-extension of H.

Let G = H + v + vv1 + ··· + vvd where vv1, . . . , vvd are length edges. Then the rigidity matrix of (G, p) has form

v A ∗ R(G, p) = , 0 R(H, p) ! where p(v) − p(v1) . A =  .  . p(v) − p(v )  d    Since p is generic, A is non-singular and hence rank R(G, p) = rank A+rank R(H, p) = d + d|V (H)| − d = d|V (G)| − d. Case 2: G is a direction-pure d-dimensional 0-extension of H.

134 Chapter 6. Inﬁnitesimal rigidity

Let G = H + v + vv1 + vv2 where vv1, vv2 are direction edges. Then the rigidity matrix of (G, p) has form

v B ∗ R(G, p) = , 0 R(H, p) ! where ⊥ [p(v) − p(v1)] B = ⊥ . [p(v) − p(v2)] ! Then since p is generic, it is easy to see that B is full column rank. Therefore rank R(G, p) ≥ rank B + rank R(H, p) = d + d|V (H)| − d = d|V (G)| − d holds.

Case 3: G is a mixed d-dimensional 0-extension of H.

Let G = H + v + vv1 + vv2 where vv1 is a length edge and vv2 is a direction edge. Let p be a generic embedding of V (G) in Rd. Then the rigidity matrix of (G, p) has form v C ∗ R(G, p) = , 0 R(H, p) ! where

p(v) − p(v1) C = ⊥ . [p(v) − p(v2)] ! It also follows from the genericity of p that C is a non-singular matrix. Therefore rank R(G, p) ≥ rank C + rank R(H, p) = d + d|V (H)| − d = d|V (G)| − d holds.

6.4.3 1-extensions for direction-length graphs

We now deﬁne the operations of 1-extensions for direction-length graphs in general dimension which extend those for 2-dimensional case introduced by Servatius and Whiteley in [94].

Deﬁnition 6.4.2. Let H be a direction-length graph and v1v2 be a length edge in H. A direction-length graph G is said to be a length-pure d-dimensional 1-extension of H on edge v1v2 if it can be obtained from H by deleting the edge v1v2, adding a new vertex v and connecting v to v1, v2 and other d − 1 vertices of H by length edges (Figure 6.5).

135 6.4. Direction-length frameworks

v •

length-pure • • ... • • • ... • v1 v2 vd+1 1-extension v1 v2 vd+1

Figure 6.5: Length-pure d-dimensional 1-extension.

v •

direction-pure • • • • • • v1 v2 v3 1-extension v1 v2 v3

Figure 6.6: Direction-pure d-dimensional 1-extension.

Deﬁnition 6.4.3. Let H be a direction-length graph and v1v2 be a direction edge in H. A direction-length graph G is said to be a direction-pure d-dimensional 1- extension of H on the edge v1v2 if it can be obtained from H by deleting the edge v1v2, adding a new vertex v and connecting v to v1, v2 and one other vertex of H by direction edges (Figure 6.6).

Deﬁnition 6.4.4. Let H be a direction-length graph and v1v2 be an edge in H (direction edge or length edge). A direction-length graph G is said to be a reduced mixed d-dimensional 1-extension of H on the edge v1v2 if it can be obtained from

H by deleting the edge v1v2, adding a new vertex v and connecting v to vertices in

H by 3 edges e1, e2, e3 such that e1 is connected to v1, e2 is connected to v2 and there are at least one direction edge and one length edge in {e1, e2, e3} (Figure 6.7).

In fact, one should expect that all the d-dimensional 1-extensions verify the property that

the number of new constraints − the number of deleted constraints ≥ d, noting that each length edge induces one linear constraint while each direction edge induces d − 1 linear constraints. However, a mixed 1-extension that veriﬁes this property does not necessarily preserve generic local rigidity. To the extent of our results in this section and in the section on global rigidity, the above deﬁned 1-extensions and reduced mixed 1-extensions are convenient and when restricted to dimension 2 they agree with the operations introduced by Servatius and Whiteley

136 Chapter 6. Inﬁnitesimal rigidity

•v •v

• • • • • • v1 v2 v3 v1 v2 v3 • • • v v v1 v2 v3 • •

reduced mixed • • • • • • 1-extension v1 v2 v3 v1 v2 v3 •v •v • • • v1 v2 v3 • • • • v1 v2 v1 v2

Figure 6.7: Reduced mixed d-dimensional 1-extension.

[94]. When there is no need to specify, we use “1-extension” to refer to any kinds of 1-extensions or reduced 1-extensions deﬁned above. The next propositions show the generic local rigidity preservingness of 1-extensions under certain conditions.

Proposition 6.4.4. Let H be a generically locally rigid direction-length graph in Rd and e be a length edge of H. Suppose that G is a d-dimensional 1-extension of H on e. Then G is generically locally rigid in Rd.

Proof. Let e = xy. We consider the following cases.

Case 1: G is a length-pure 1-extension of H.

Let G = H − xy + vx + vy + vz1 + ··· vzd−1. Let p be an embeding of G such that the restriction of p on V (H) is generic, p(v) is the midpoint of the segment

[p(x), p(y)], and p(v) is not in a hyperplane containing p(y), p(z1), . . . , p(zd−1). The three rows of R(G + xy, p) indexed by xy, vx, vy then form a circuit in the linear matroid deﬁned on the rows of this matrix. Therefore,

rank R(G, p) = rank R(G + xy, p) = rank R(G + xy − vx, p).

Matrix R(G + xy − vx, p) has form

v A ∗ , 0 R(H, p) !

137 6.4. Direction-length frameworks where p(v) − p(y)

p(v) − p(z1) A =   . .  .    p(v) − p(zd−1)   Since the restriction of p on V (H) is generic and p(v) is not in a hyperplane containing p(y), p(z1), . . . , p(zd−1), A is non-singular. Therefore, rank R(G, p) = rank R(G + xy − vx, p) = rank A + rank R(H, p) = d + d|V (H)| − d = d|V (G)| − d, since (H, p) is locally rigid. For any generic embedding q of G, rank R(G, q) ≥ rank R(G, p) = d|V (G)| − d holds. Therefore, G is generically locally rigid by Theorem 6.4.1. Case 2: G is a reduced mixed 1-extension of H. Let G = H + vx + vy + vz − xy. Case 2.1: vx, vy are length-edges and vz is a direction edge. Taking an embedding p as in Case 1, we have,

rank R(G, p) = rank R(G + xy, p) = rank R(G + xy − vx, p), and v B ∗ R(G + xy − vx, p) = , 0 R(H, p) ! where p(v) − p(y) B = ⊥ . [p(v) − p(z)] ! is a non-singular matrix as the restriction of p on V (H) is generic and p(v) = 1 2 [p(x) + p(y)]. Then rank R(G, p) = rank B + rank R(H, p) = d|V (G)| − d holds. That implies G is generically locally rigid according to Theorem 6.4.1. Case 2.2: vx, vy are direction edges and vz is a length edge. Consider an embedding p of G which is generic on V (H). Let w be a unit vector in Rd such that w is perpendicular to p(x) − p(y) but not perpendicular to p(v) − p(z). The subspace of Rd orthogonal to w contains p(x) − p(y), so we can take w⊥ with the ﬁrst row being (p(x) − p(y))/kp(x) − p(y)k. Let R∞(G + xy, p),R∞(G + xy − vx, p) be the matrices obtained from R(G + xy, p),R(G + xy − vx, p) respectively by replacing [p(v) − p(x)]⊥,[p(v) − p(y)]⊥, and p(v) − p(z) with w. Let R∞(G, p) denote the matrix obtained from R∞(G + xy, p) by deleting

138 Chapter 6. Inﬁnitesimal rigidity

w v • z • w •v x••y • •• z x y (a) (b)

Figure 6.8: (a) v tends to infinity in the direction of w. (b) v tends to y in the direction of w. the row indexed by xy. We can view R∞(G, p) as the limit of R(G, p) (up to a non-zero scalar scaling of some rows) when p goes to infinity in the direction of w (Figure 6.8(a)). In the linear matrix defined on the rows of R∞(G + xy, p), by the choice of w⊥, the first row of the submatrix indexed by vx, the first row of the submatrix indexed by vy and the row indexed by xy form a circuit. This implies rank R∞(G, p) = rank R∞(G + xy, p) = rank R∞(G + xy − vx, p). We have,

v C ∗ R∞(G + xy − vx, p) = , 0 R(H, p) ! where w C = ⊥ w ! is a non-singular matrix. Therefore, rank R∞(G+xy −vx, p) ≥ rank R(H, p)+d = d|V (G)| − d, since H is generically locally rigid. It follows that rank R∞(G, p) ≥ d|V (G)| − d. Since the rank function of a matrix is lower semi-continuous, there exists an embedding p of G such that rank R(G, p) ≥ rank R∞(G, p) ≥ d|V (G)|−d. Therefore, G is generically locally rigid by Theorem 6.4.1. Case 2.3: vx is a length edge, vy, vz are direction edges. Consider p, w and w⊥ as in Case 2.2. Let R◦(G, p),R◦(G + xy, p) and R◦(G + xy − vx) be respectively obtained from R(G, p),R(G + xy, p),R(G + xy − vx, p) by replacing p(v) −p(x) with p(y) −p(x), [p(v) −p(y)]⊥ with w⊥ and [p(v) −p(z)]⊥ with [p(y) − p(z)]⊥. We can view R◦(G, p) as the limit of R(G, p) when p(v) tends to p(y) in the direction of w (Figure 6.8(b)). In the linear matroid deﬁned on the rows of R◦(G + xy, p), the rows corresponding to vx, xy and the ﬁrst row

139 6.4. Direction-length frameworks of the submatrix corresponding to vy form a circuit. Therefore, rank R◦(G, p) = rank R◦(G + xy, p) = rank R◦(G + xy − vx, p). We have

v D ∗ R◦(G + xy − vx, p) = , 0 R(H, p) ! where [p(y) − p(z)]⊥ D = ⊥ w ! is a full column rank matrix. Therefore, rank R◦(G + xy −vx, p) ≥ rank R(H, p) + d = d|V (G)| − d, since H is generically locally rigid. This implies rank R(G, p) ≥ d|V (G)| − d for some p with p(v) close enough to p(y). It follows from Theorem 6.4.1 that G is generically locally rigid. Case 2.4: vx, vz are length edges and vy is a direction edge. The proof is similar to that of Case 2.3.

1-extensions on a direction edge are more complicated as the deletion of a direction edge can reduce the rank of the rigidity matrix up to d − 1. However, if the rank decreases by at most 1, an argument similar to the one above gives the following.

Proposition 6.4.5. Let e be a direction edge of a generically locally rigid direction- length graph H and G be a d-dimensional 1-extension of H on e. Suppose that the deletion of e decreases the rank of the rigidity matrix of a generic realization of H by at most 1. Then G is generically locally rigid in Rd.

Without the condition on the decrease of the rank, we have the next result for 1-extensions with two new direction edges incident to the vertices of the deleted edge.

Proposition 6.4.6. Let H be a generically locally rigid direction-length graph in Rd and xy be a direction edge of H. Suppose that G = H − xy + vx + vy + vz is a 1-extension of H on xy where vx, vy are direction edges. Then G is generically locally rigid in Rd.

Proof. Let p be an embedding of G such that p is generic on V (H) and p(v) is the midpoint of the segment [p(x), p(y)]. We can take [p(v)−p(x)]⊥ = [p(v)−p(y)]⊥ = [p(x) − p(y)]⊥. Then in the linear matroid deﬁned on the rows of R(G + xy, p),

140 Chapter 6. Inﬁnitesimal rigidity the ith rows of the submatrices indexed by vx, vy and xy form a circuit, for each i = 1, . . . , n. Hence, rank R(G, p) = rank R(G + xy, p) = rank R(G − vx + xy, p). As in the proof of Proposition 6.4.4, it is not diﬃcult to show that this rank is equal to rank R(H, p) + d = d|V (G)| − d since H is generically locally rigid. Therefore, G is generically locally rigid.

6.4.4 Union of two spanning trees

In this section we use the results in Section 6.4.2 and 6.4.3 to show the generic local rigidity in all dimensions of a special class of direction-length graphs.

Theorem 6.4.7. Let G = (V,D ∪ L) be a direction-length graph such that both (V,D) and (V,L) are spanning trees. Then G is generically locally rigid in all dimensions.

Proof. We prove the theorem by induction on the number of vertices |V |. The theorem is trivial when |V | = 2. Suppose that the theorem holds for every direction- length graph with at most |V | − 1 vertices. Since G is the union of two spanning trees, a simple count shows that there is a vertex v of G of degree 2 or 3.

Case 1: G has a vertex v of degree 2. Then it is obvious that G − v is the union of a length spanning tree and a direction spanning tree on V −v. Thus G−v is generically locally rigid by induction hypothesis. The direction-length graph G is obtained from G − v by adding a vertex v with a length edge and a direction edge connecting v to V − v, that is, a d-dimensional 0-extension, for every d. Therefore, by Proposition 6.4.3 G is generically locally rigid in Rd for every d ≥ 1.

Case 2: G has a vertex v of degree 3 and v is incident to two length edges vv1, vv2 and a direction edge vv3.

As L is the edge set of a spanning tree and vv1, vv2 ∈ L, the length edge e = v1v2 is not in L. Then L−vv1 −vv2 +e is the edge set of a length spanning tree on V −v.

We also have that D −vv2 is the edge set of a direction spanning tree on V −v. By induction, H = G−v +e is a generically locally rigid direction-length graph. Since G is a d-dimensional 1-extension of H on a length edge, G is generically locally rigid by Proposition 6.4.4.

Case 3: G has a vertex v of degree 3 and v is incident to two direction edges vv1, vv2 and a length edge vv3.

141 6.4. Direction-length frameworks

As D is the edge set of a spanning tree and vv1, vv2 ∈ D, the direction edge e = v1v2 is not in D. Then D −vv1 −vv2 + e is the edge set of a direction spanning tree on V − v. We also have L − vv3 is the edge set of a spanning tree on V − v. By induction, H = G − v + e is a generically locally rigid direction-length graph. Since G is a d-dimensional 1-extension of H on a direction edge e with new vertex v and two direction edges vv1, vv2 incident to the vertices of e, G is generically locally rigid by Proposition 6.4.6.

An immediate consequence of this theorem is that if a direction-length graph G contains a spanning direction tree and a spanning length tree then it is generically locally rigid in Rd for every d.

142 Chapter 7

Global rigidity of direction-length frameworks

Contents 7.1 Introduction ...... 144 7.2 Quasi-generic direction-length frameworks ...... 145 7.3 Proof of Theorem 7.1.2 ...... 147 7.4 Further remarks ...... 156

143 7.1. Introduction

7.1 Introduction

In Section 6.4 we have investigated the effect of 1-extensions on the local rigidity of direction-length frameworks. This chapter is devoted to the study of the effect of 1-extensions on the global rigidity of generic direction-length frameworks. A direction-length framework (G, p) is globally rigid if every direction-length framework equivalent to (G, p) is congruent to (G, p). In the sequel of this chapter, we will restrict our attention to algebraically generic direction-length frameworks, which we will call generic for short. A framework (G, p) is (algebraically) generic if the set containing all coordinates of its vertices is algebraically independent over the rationals. A direction-length graph G is generically globally rigid in Rd if every algebraically generic realization of G in Rd is globally rigid. As we have seen in Section 6.4, the local rigidity of direction-length frameworks is a generic property (Theorem 6.4.1). Meanwhile, whether global rigidity of direction-length frameworks is a generic property or not is an open problem even in dimension 2. For direction frameworks, where only directions are of interest, it is known that both local rigidity and global rigidity are generic properties [109]. Fur- thermore, a direction graph is generically globally rigid if and only if it is generically locally rigid. The question if global rigidity is a generic property for bar-and-joint frameworks is answered affirmatively for general dimension by Gortler, Healy and Thurston [44] with an algebraic geometry approach and, earlier, for dimension 2, by Jackson and Jordán [58] with a combinatorial constructive proof. The proof of Gortler, Healy and Thurston employed stress matrices for bar-and-joint frameworks, a notion that seems to be difficult to extend to direction-length frameworks, while the constructive proof by Jackson and Jordán is more likely extendable. In fact, Jackson and Jordán showed that the underlying graph of a globally rigid

2-dimensional generic bar-and-joint framework can be contructed from K4 by a sequence of 1-extensions and edge-additions, two operations preserving the generic global rigidity of graphs. In [61], they explored the global rigidity preservingness of 1-extension for direction-length frameworks in R2. In this chapter, we extend the result of Jackson and Jord´an in [61] to general dimension. Recall from Section 6.4 that a d-dimensional length-pure 1-extension of a direction- length H deletes a length edge v1v2 of H, adds a new vertex v then connects v to v1, v2 and d − 1 other vertices of H by length edges (Figure 6.5). A d-dimensional direction-pure 1-extension of H deletes one direction edge v1v2 of H, adds a new

144 Chapter 7. Global rigidity of direction-length frameworks

vertex v then connects v to v1, v2 and one other vertex of H by direction edges (Figure 6.6). A d-dimensional reduced mixed 1-extension of H the operation that deletes an edge v1v2 of H, adds a new vertex v then connects v to vertices in H by 3 edges e1, e2, e3 such that e1 is incident to v1, e2 is incident to v2 and there is at least one direction edge and one length edge in {e1, e2, e3} (Figure 6.7). When restricted to dimension 2 these operations agree with 1-extensions for direction-length frameworks introduced by Servatius and Whiteley [94]. Jackson and Jord´an proved the following result.

Theorem 7.1.1 ([61]). Let H be a direction-length graph with |V (H)| ≥ 3 and G a direction-length graph obtained from H by a 2-dimensional 1-extension on an edge e. Suppose that H is generically globally rigid and H − e is rigid in R2. Then G is generically globally rigid in R2.

Contribution: In using the same algebraic approach as Jackson and Jord´an in [61] with a modiﬁed technique, we extend their above result to all dimensions. Our technique results in a simpler calculation in the proof for the case d = 2.

Theorem 7.1.2. Let H be a direction-length graph with |V (H)| ≥ d + 1 and G a direction-length graph obtained from H by either a d-dimensional length-pure 1- extension, direction-pure 1-extension or a reduced mixed 1-extension on an edge e. Suppose that H is generically globally rigid and H − e is generically locally rigid in Rd. Then G is generically globally rigid in Rd.

This result provides a tool to verify the generic global rigidity of certain direction- length graphs. It can be also used for generating families of generically globally rigid direction-length graphs.

The results in this chapter are from our paper [86].

7.2 Quasi-generic direction-length frameworks

To prepare for the proof of our main theorem, in this section, we introduce and summarize some results on quasi-generic direction-length frameworks. A quasi- generic direction-length framework is simply a translation of a generic direction- length framework. It is convenient to work with a quasi-generic direction-length framework with one vertex coinciding with the origin. The following lemma is trivial.

145 7.2. Quasi-generic direction-length frameworks

Lemma 7.2.1. Let (G, p) be a direction-length framework with V = V (G) =

{v0, v1, . . . , vn}, p(v0) = (0,..., 0) and p(vi) = (pd(i−1)+1, . . . , pdi) for 1 ≤ i ≤ n. Then (G, p) is quasi-generic if and only if {p1, p2, . . . , pdn} is algebraically independent over the rationals.

For a point p ∈ Rn, we use Q(p) to denote the field extension of the rational field Q by coordinates of p. Let K be a field. We denote by K the algebraic closure of K. We will need the following lemmas.

Lemma 7.2.2 ([61]). Let fi, gi be non-zero polynomials in n variables with rational n coeﬃcients, for 1 ≤ i ≤ m. Let Ti = {x ∈ R : gi(x) =6 0} for 1 ≤ i ≤ m and put m m T = i=1 Ti. Let f : T → R deﬁned by f(x) = (r1(x), r2(x), . . . , rm(x)) where r (x) = f (x)/g (x) for 1 ≤ i ≤ m. i T i i

n 1. Suppose that maxx∈Rn {rank df|x} = m. If p is a generic point in R then p ∈ T and f(p) is a generic point in Rm.

2. Suppose that m = n and f(p) is a generic point in Rn for some p ∈ T . Then p is generic and Q(p) = Q(f(p)).

Let (G, p) be a direction-length framework in Rd, where G = (V,D ∪ L) and

D ∪ L = {e1, e2, . . . , em}. We choose a reference orientation for edges in G. Let lp : L → R deﬁned by

2 lp(e) = kp(u) − p(v)k for e = uv ∈ L.

d−1 Let dp : D → R be deﬁned by

p (u) − p (v) p (u) − p (v) d (e) = 1 1 ,..., d−1 d−1 if e = uv ∈ D, p p (u) − p (v) p (u) − p (v) d d d d where pi(u) denotes the ith coordinate of p(u) for every 1 ≤ i ≤ d, assuming that this map is well-deﬁned. Let rp(e) = lp(e) if e ∈ L and rp(e) = dp(e) if e ∈ D.

Putting all the components of rp(e1), rp(e2), . . . , rp(em) consecutively to obtain a vector fG(p), we call fG the rigidity map of the direction-length graph G. Then it is easy to see that two realizations (G, p) and (G, q) of G are equivalent if and only if fG(p) = fG(q), assuming that the rigidity map is well-deﬁned at both p and q. It is also known that a generic direction-length framework (G, p) is locally rigid if and only if rank dfG|p = d|V | − d (see [64, Lemma 8.1]).

146 Chapter 7. Global rigidity of direction-length frameworks

Lemma 7.2.3. Let (G, p) be a locally rigid quasi-generic direction-length frame- d work in R with V = {v0, v1, . . . , vn}, p(v0) = (0, 0,..., 0), and p(vi) = (pd(i−1)+1, pd(i−1)+2, . . . , pdi) for 1 ≤ i ≤ n. Then (p1, p2, . . . , pdn) is generic and Q(p) =

Q(fG(p)). Proof. Assume that (G, p) is locally rigid and is a translation of a generic direction- length framework (G, q). Then (G, q) is locally rigid, so rank dfG|q = d|V | − d. ′ ′ Therefore, there exists a projection fG of fG such that fG has d|V | −d components ′ ′ and rank dfG|q = d|V | − d. From Lemma 7.2.2 (1) we have that fG(q) is generic, ′ thus fG(p) is generic. Applying Lemma 7.2.2 (2), we have that (p1, p2, . . . , pdn) is generic and Q(p) = Q(fG(p)).

The following corollary of Lemma 7.2.3 is crucial in the proof of our main theorem.

Corollary 7.2.4. Suppose that (G, p) is a locally rigid quasi-generic direction- length framework, (G, q) is equivalent to (G, p), and p(v0) = q(v0) = (0, 0,..., 0). Then (G, q) is quasi-generic and Q(p) = Q(q).

7.3 Proof of Theorem 7.1.2

Let (G, p) be a generic direction-length framework and u, v ∈ V (G). The pair {u, v} is said to be globally distance linked (globally direction linked, resp.) in

(G, p) if lp(uv) = lq(uv)(dp(uv) = dq(uv), resp.) for every framework (G, q) equivalent to (G, p). The pair {u, v} is said to be globally linked in (G, p) if it is both globally direction linked and globally distance linked in (G, p). The pair {u, v} is generically globally distance linked (generically globally direction linked or generically globally linked, resp.) in G if it is globally distance linked (globally direction linked or globally linked, resp.) in every generic realization of G. Theorem 7.1.2 is just a corollary of the following theorem.

Theorem 7.3.1. Let H be a direction-length graph with |V (H)| ≥ d + 1 and G be a d-dimensional 1-extension of H with new vertex v. Suppose that G − v is generically locally rigid.

(a) If G is a length-pure 1-extension of H then every pair of neighbors of v is generically globally distance linked in G.

(b) If G is a direction-pure 1-extension of H then every pair of neighbors of v is generically globally direction linked in G.

147 7.3. Proof of Theorem 7.1.2

Proof. Let V = V (G) = {v0, v1, . . . , vn} and suppose that v = vn. Let (G, p) be a quasi-generic realization of G and (G, q) an equivalent realization of (G, p). Assume that

p(v0) = q(v0) = (0, 0,..., 0),

p(vi) = (pd(i−1)+1, pd(i−1)+2, . . . , pdi) for 1 ≤ i ≤ n,

q(vi) = (qd(i−1)+1, qd(i−1)+2, . . . , qdi) for 1 ≤ i ≤ n.

By Lemma 7.2.1, {p1, p2, . . . , pn} is algebraically independent over Q. ′ ′ Let p = p|V −v and q = q|V −v, the restriction of p and q on the vertex set of H. Then (G − v, q′) is equivalent to (G − v, p′) and (G − v, p′) is quasi-generic. Since G − v is generically locally rigid, Corollary 7.2.4 implies that (G − v, q′) ′ ′ is quasi-generic and Q(p ) = Q(q ). In particular, the set {q1, q2, . . . , qd(n−1)} is algebraically independent over the rationals. Let K = Q(p′) = Q(q′).

Proof of (a)

Without loss of generality, we may assume that the set of neighbors of v is {v0, v1, . . . , vd}. Since (G, p) and (G, q) are equivalent, we have the following equations

2 2 2 2 qd(n−1)+1 + ··· + qdn = pd(n−1)+1 + ··· + pdn, (0) 2 2 2 (qd(n−1)+1 − q1) + ··· + (qdn − qd) = (pd(n−1)+1 − p1) 2 + ··· + (pdn − pd) , (1) . . 2 2 2 (qd(n−1)+1 − qd(d−1)+1) + ··· + (qdn − qd2 ) = (pd(n−1)+1 − pd(d−1)+1) 2 + ··· + (pdn − pd2 ) . (d) Subtracting equations (1), . . . , (d) from equation (0), we have the following system of linear equations:

T T Aqq(vn) = App(vn) + a, where Ap,Aq are d × d matrices with all components belong to K and a is a vector with d components in K. In fact,

q1 q2 ··· qd  qd+1 qd+2 ··· q2d A = q . . .. .  . . . .    qd(d−1)+1 qd(d−1)+2 ··· qd2      148 Chapter 7. Global rigidity of direction-length frameworks

is a non-singular matrix since {q1, q2, . . . , qd(n−1)} is algebraically independent over the rationals, and 2 2 kq(v1)k − kp(v1)k 2 2 1 kq(v2)k − kp(v2)k  a = × . 2 .  .   2 2 kq(vd)k − kp(vd)k    T −1  T −1  Therefore, q(vn) = Aq App(vn) + Aq a. In particular, qd(n−1)+1, . . . , qdn ∈

K(pd(n−1)+1, . . . , pdn).

Since (0) holds, the constant term of each qj, d(n − 1) + 1 ≤ j ≤ dn, regarded −1 as polynomial of pd(n−1)+1, . . . , pdn in K, is 0. Thus, Aq a = 0, which implies that a = 0. Therefore {v0, v1}, {v0, v2},..., {v0, vd} are globally distance linked in G. Symmetry implies that all pairs of neighbors of v are globally distance linked in G.

Proof of (b)

Without loss of generality we can assume that the set of neighbors of v is {v0, v1, v2}. Since (G, p) and (G, q) are equivalent, we have the following systems of equations q p dn−1 = dn−1 , qdn pdn q − q p − p dn−1 d−1 = dn−1 d−1 , qdn − qd pdn − pd q − q p − p dn−1 2d−1 = dn−1 2d−1 , qdn − q2d pdn − p2d which is equivalent to

qdn−1pdn = qdnpdn−1, (i)

(qdn−1 − qd−1)(pdn − pd) = (qdn − qd)(pdn−1 − pd−1), (ii)

(qdn−1 − q2d−1)(pdn − p2d) = (qdn − q2d)(pdn−1 − p2d−1). (iii)

Subtracting equations (ii) and (iii) from (i), we obtain the following linear system of equations

T T A(qdn−1, qdn) = B(pdn−1, pdn) + c, where p −p A = d d−1 p2d −p2d−1!

149 7.3. Proof of Theorem 7.1.2 is a non-singular matrix, and q p − q p c = d−1 d d d−1 . q2d−1p2d − q2dp2d−1! T −1 T −1 Therefore, (qdn−1, qdn) = A B(pdn−1, pdn) + A c. Substituting this into (i), we obtain a polynomial of pdn, pdn−1 over K which must be identically zero. It is easy −1 to see that the coefficient of pdn in this polynomial is the first component of A c and the coefficient of pdn−1 is equal to the negative of the second component of A−1c. Thus A−1c must be zero, which implies that c = 0. Therefore, q p d−1 = d−1 , qd pd and q p 2d−1 = 2d−1 q2d p2d hold. By symmetry we obtain similar equalities for other components and other pairs of neighbors of v. Therefore, every pair of neighbors of v is globally linked in G.

Proof of (c)

(c-1) We consider the case that the mixed 1-extension has one length edge vnv0 and two direction edges vnv1, vnv2. Using the fact that (G, p) and (G, q) are equivalent, we have

2 2 2 2 2 2 qd(n−1)+1 + qd(n−1)+2 + ··· + qdn = pd(n−1)+1 + pd(n−1)+2 + ··· + pdn, (0’)

(qdn − qd)(pd(n−1)+1 − p1) = (qd(n−1)+1 − q1)(pdn − pd), (1’) . .

(qdn − qd)(pdn−1 − pd−1) = (qdn−1 − qd−1)(pdn − pd), ((d-1)’)

(qdn − q2d)(pdn−1 − p2d−1) = (qdn−1 − q2d−1)(pdn − p2d). (d’) We rewrite the system of equations from (1’) to (d’) as

T T Xq(vn) = Y p(vn) + z, where pdn − pd 0 −(pd(n−1)+1 − p1) . .  .. .  X = ,  0 pdn − pd −(pdn−1 − pd−1)     pdn − p2d −(pdn−1 − p2d−1)     150 Chapter 7. Global rigidity of direction-length frameworks and p1qd − pdq1 .  .  z = .  pd−1qd − pdqd−1    p2d−1q2d − p2dq2d−1     Let Xi be the matrix obtained from X by replacing the i-th column of X with T Y p(vn) + z. Let D = det X, Di = det Xi for 1 ≤ i ≤ d. Then qd(n−1)+i = Di/D for 1 ≤ i ≤ d by Cramer’s rule. Substituting into (0’), we have

2 2 2 2 2 D1 + ··· + Dd = D (pd(n−1)+1 + ··· + pdn)(∗) with D,Di ∈ K(pd(n−1)+1, . . . , pdn−1, pdn) for 1 ≤ i ≤ d.

The constant term in the right side of (∗) is zero, thus the constant term αi of each Di must be zero. Let

−pd 0 p1 . .  .. .  T =  0 −pd pd−1     −p2d p2d−1     be the matrix obtained from X by setting all pd(n−1)+1, . . . , pdn−1, pdn to 0 (we can consider T as the matrix consisting of the constant terms of X in K). Let Ti be the matrix obtained from T by replacing the i-th column of T with z (equivalently, Ti is the matrix consisting of the constant terms of Xi). Then det Ti = 0 for 1 ≤ i ≤ d. d−2 Note that det T = (−pd) (pd−1p2d − pdp2d−1) =6 0, Cramer’s rule implies that x = (0,..., 0) is the only solution of the linear system T x = z. Hence, z = 0 holds. Therefore, q q q d = d−1 = ··· = 1 = α. (c1) pd pd−1 p1 Similarly, we obtain q q q 2d = 2d−1 = ··· = d+1 = β. (c2) p2d p2d−1 pd+1 Since z = 0, we have

q1pdn − qdpd(n−1)+1 . T  .  Y p(vn) + z = .  qd−1pdn − qdpdn−1    q2d−1pdn − q2dpdn−1     151 7.3. Proof of Theorem 7.1.2

t1 td For a polynomial P in K[X1,...,Xd], let CoeffP (X1 ...Xd ) denote the coef- t1 td 2d ficient of the monomial X1 ...Xd in P . Then the coefficient of pdn in the left of (∗) is equal to

2d d 2 Coeﬀ 2 (p ) = [CoeﬀD (p )] Dd dn d dn 2 = (qd−1 − q2d−1)

2d and the coeﬃcient of pdn in the right of (∗) is equal to

2d−2 d−1 2 CoeﬀD2 (pdn ) = [CoeﬀD(pdn )] 2 = (pd−1 − p2d−1) .

2 2 Comparing these two coeﬃcients, we obtain that (qd−1 − q2d−1) = (pd−1 − p2d−1) 2 holds. Since the role of every component is equivalent, we have (qd − q2d) = 2 (pd − p2d) . Thus

qd−1 − q2d−1 = ±(pd−1 − p2d−1), (c3) and

qd − q2d = ±(pd − p2d). (c4) 2d−1 We now evaluate the coeﬃcient of pdn pdn−1 in the left of (∗) which is equal to

2d−1 d d−1 Coeff 2 (p pdn ) = 2 × CoeffD (p ) × CoeffD (p pdn ) Dd dn −1 d dn d dn −1

= −2(qd−1 − q2d−1)(qd − q2d),

2d−1 and the coeﬃcient of pdn pdn−1 in the right of (∗) which is equal to

2d−3 d−1 d−2 CoeffD2 (pdn pdn−1) = 2 × CoeffD(pdn ) × CoeffD(pdn pdn−1)

= −2(pd−1 − p2d−1)(pd − p2d).

Comparing these two coeﬃcients, we have (qd−1 − q2d−1)(qd − q2d) = (pd−1 − p2d−1)(pd − p2d). Together with (c3) and (c4) this equation implies that, if qd−1 − q2d−1 = pd−1 − p2d−1 then qd − q2d = pd − p2d, and if qd−1 − q2d−1 = −(pd−1 − p2d−1) then qd − q2d = −(pd − p2d). Suppose that qd−1 − q2d−1 = pd−1 − p2d−1 and qd − q2d = pd − p2d. Using (c1) and (c2) we have,

αpd−1 − βp2d−1 = pd−1 − p2d−1,

αpd − βp2d = pd − p2d.

Since {pd−1, pd, p2d−1, p2d} is algebraically independent over the rationals, we obtain α = β = 1, which implies that p(v1) = q(v1) and p(v2) = p(v1), i.e., the pairs {v0, v1}, {v0, v2}, {v1, v2} are globally linked in G.

152 Chapter 7. Global rigidity of direction-length frameworks

Similar argument can be applied for the case qd−1 − q2d−1 = −(pd−1 − p2d−1) and qd − q2d = −(pd − p2d) to deduce that p(v1) = −q(v1) and p(v2 = −q(v2). The special case that v has just two neighbors in G − v, i.e., v is connected to v0 by one length edge and one direction edge, and v is connected to v1 by one length edge can be treated similarly as above by putting q(v2) = p(v2) = 0. (c-2) We now consider the case that the mixed 1-extension has one direction edge vnv0 and two length edges vnv1, vnv2. Since (G, p) and (G, q) are equivalent we have the following equations q p d(n−1)+1 = d(n−1)+1 , qdn pdn . . q p dn−1 = dn−1 , qdn pdn 2 2 2 2 (qd(n−1)+1 − q1) + ··· + (qdn − qd) = (pd(n−1)+1 − p1) + ··· + (pdn − pd) , 2 2 2 2 (qd(n−1)+1 − qd+1) + ··· + (qdn − q2d) = (pd(n−1)+1 − pd+1) + ··· + (pdn − p2d) .

From the two last equations we derive a linear equation of q(vn). Together with the ﬁrst d − 1 equations we obtain,

T Mq(vn) = r, where pdn 0 −pd(n−1)+1 . .  .. .  M = ,  0 pdn −pdn−1    q1 − qd+1 . . . qd−1 − q2d−1 qd − q2d      0 r = , s + r! with

s = (p1 − pd+1)pd(n−1)+1 + ··· + (pd − p2d)pdn, and 1 r = (kq(v )k2 − kp(v )k2 − kq(v )k2 + kp(v )k2). 2 1 1 2 2 Let Mi be the matrix obtained from M by replacing the i-th column of M with d−2 r. Let F = det M and Fi = det Mi for 1 ≤ i ≤ d. Then Fi = (s + r)pdn pd(n−1)+i d−1 for 1 ≤ i ≤ d. In particular, Fd = (s + r)pdn . We can observe that the highest degree of pdn in F , considered as a polynomial over K, is d − 1.

153 7.3. Proof of Theorem 7.1.2

Solving the linear system for q(vn) we obtain qd(n−1)+i = Fi/F for i = 1, . . . , d.

We now rewrite the equation for vnv1 as

2 2 2 2 2 (F1 − q1F ) + ··· + (Fd − qdF ) = F [(pd(n−1)+1 − p1) + ··· + (pdn − pd) ]. (∗∗)

2d We ﬁrst evaluate the coeﬃcient of pdn in the left of (∗∗) which is equal to

2d d 2 Coeﬀ 2 (p ) = [CoeﬀF (p )] Fd dn d dn 2 = (pd − p2d) ,

2d and the coeﬃcient of pdn in the right of (∗∗) which is equal to

2d−2 d−1 2 CoeﬀF 2 (pdn ) = [CoeﬀF (pdn )] 2 = (qd − q2d) .

Thus, by comparing these two coeﬃcients,

2 2 (pd − p2d) = (qd − q2d) (c5) holds. 2d−2 We then evaluate the coeﬃcient of pdn in the left of (∗∗), which is

2d−2 2 2 2d−2 d−1 d−1 Coeff 2 (p ) + (q + ··· + q ) × Coeff 2 (p ) − 2q × Coeff (p ) × Coeff (p ) Fd dn 1 d F dn d Fd dn F dn 2 2 2 = r + kq(v1)k (qd − q2d) − 2qdr(qd − q2d),

2d−2 and the coeﬃcient of pdn in the right of (∗∗)

2 2 2d−2 2 2 (p1 + ··· + pd)CoeﬀF 2 (pdn ) = kp(v1)k (qd − q2d) .

It follows that

2 2 2 2 r − 2qdr(qd − q2d) = (kp(v1)k − kq(v1)k )(qd − q2d) . (c6)

2d−1 We also evaluate the coeﬃcient of pdn in the left of (∗∗) which is

d d−1 2d−1 2 −2qd × CoeffFd (pdn) × CoeffF (pdn ) + CoeffF (pdn )

= −2qd(pd − p2d)(qd − q2d) + 2r(pd − p2d),

2d−1 and the coeﬃcient of pdn in the right of (∗∗) which is

2d−1 2 −2pd × CoeﬀF 2 (pdn ) = −2pd(qd − q2d) 2 = −2pd(pd − p2d) (by (c5)).

154 Chapter 7. Global rigidity of direction-length frameworks

2 Therefore, −2qd(pd − p2d)(qd − q2d) + 2r(pd − p2d) = −2pd(pd − p2d) , which is equivalent to r − qd(qd − q2d) = −pd(pd − p2d). Squaring both sides of this 2 2 2 2 2 equation, we have r − 2rqd(qd − q2d) + qd(qd − q2d) = pd(pd − p2d) . Using (c6) 2 2 2 2 2 2 2 we have (kp(v1)k − kq(v1)k )(qd − q2d) + qd(qd − q2d) = pd(pd − p2d) . Since 2 2 2 2 2 2 (qd −q2d) = (pd −p2d) by (c5), this implies that kp(v1)k −kq(v1)k = pd −qd. This equation must hold for every coordinate due to the equivalence of their roles. It 2 2 2 2 2 2 2 2 follows that d(kp(v1)k −kq(v1)k ) = (p1 −q1 )+···+(pd −qd ) = kp(v1)k −kq(v1)k . 2 2 2 2 Therefore kp(v1)k = kq(v1)k holds. Similarly, kp(v2)k = kq(v2)k holds. Thus r = 0 holds, then qd(qd − q2d) = pd(pd − p2d) holds. Together with (c5), this implies that p p − p p d = d 2d = 2d = γ = ±1 (c7) qd qd − q2d q2d holds. Similarly, p p − p p d−1 = d−1 2d−1 = 2d−1 = δ = ±1 (c8) qd−1 qd−1 − q2d−1 q2d−1 holds. 2d−1 Last, we evaluate the coeﬃcient of pdn pdn−1 in the left of (∗∗)

2d−1 d d−1 Coeff 2 (p p ) = 2 × Coeff (p ) × Coeff (p p ) Fd dn dn−1 Fd dn Fd dn dn−1

= 2(pd − p2d)(pd−1 − p2d−1),

2d−1 and the coeﬃcient of pdn pdn−1 in the right of (∗∗) which is

2d−3 d−1 d−2 CoeffF 2 (pdn pdn−1) = 2 × CoeffF (pdn ) × CoeffF (pdn pdn−1)

= 2(qd − q2d)(qd−1 − q2d−1).

Thus, by comparing these two coeﬃcients, (pd − p2d)(pd−1 − p2d−1) = (qd − q2d)(qd−1 − q2d−1) holds. Together with (c7) and (c8) we have γ = δ. By symmetry we conclude that {v0, v1}, {v0, v2} and {v1, v2} are globally linked in G. The special case that v has just 2 neighbors in G − v, i.e., v is connected to v0 by one direction edge and one length edge, and v is connected to v1 by one direction edge, can be treated similarly as above by putting q(v2) = p(v2) = 0.

Proof of Theorem 7.1.2. Let (G, p) be a generic realization of G in Rd and (G, q) a realization in Rd that is equivalent to (G, p). We show that (G, q) is congruent to (G, p). Let e = v1v2 and v be the new vertex. Since G−v = H −e is generically locally rigid, applying Theorem 7.3.1, we have that, in (G, p), the pair {v1, v2} is globally direction linked if e is a direction edge and {v1, v2} is globally distance linked if e is a length edge. Therefore, (H, q) is equivalent to (H, p) and hence

155 7.4. Further remarks they are congruent as H is generically globally rigid. So there exists a realization (G, p′) congruent to (G, p) and a realization (G, q′) congruent to (G, q) such that (H, q′) coincides to (H, p′). Then it is not diﬃcult to see that the distance and/or direction constraints between v and its neighbors uniquely determine the position of v, which means that p′(v) = q′(v). Therefore, (G, p′) is congruent to (G, q′), which implies that (G, p) is congruent to (G, q). 2

7.4 Further remarks

The assumption about the rigidity of H − e is essential in our proof. This assumption is however not necessary for the analogous result to Theorem 7.1.2 for bar-and-joint frameworks [62] due to the fact that a globally rigid generic bar-and- joint framework is redundantly locally rigid [53]. The behavior of a globally rigid direction-length framework after deleting an edge is much more complicated. Some partial results on this behavior have been obtained by Jackson and Keevash.

Theorem 7.4.1 ([65]). Suppose that G is a generically globally rigid direction- length graph in Rd with at least two length edges. Then G − e is generically locally rigid in Rd for any length edge e of G.

Combining with Theorem 7.1.2, we obtain the following consequence.

Corollary 7.4.2. Suppose that H is a globally rigid direction-length graph in Rd with at least two length edges and G is obtained from H by a d-dimensional 1- extension on a length edge. Then G is generically globally rigid in Rd.

The suppression of a direction edge removes d − 1 constraints. Therefore, the behavior of a direction-length framework after deleting a direction edge must be much more complicated in dimension d ≥ 3. Even in dimension 2, only partial results are obtained. A direction-length framework (G, p) is said to be bounded if there exists a K > 0 such that kq(u) − q(v)k < K for every framework (G, q) equivalent to (G, p). Jackson and Keevash showed the following.

Theorem 7.4.3 ([65]). Suppose that G is a 2-dimensional generically globally rigid direction-length graph, e is a direction edge of G, and G−e has a generically locally rigid subgraph with more than one vertex. Then G − e is either rigid or unbounded in R2.

156 Chapter 7. Global rigidity of direction-length frameworks

They conjectured that the assumption about the existence of a rigid subgraph with more than one vertex of G − e is not necessary.

Conjecture 7.4.4 ([64]). Suppose that G is a 2-dimensional generic globally rigid direction-length framework with at least two length edges and e is a direction edge of G. Then G − e is either rigid or unbounded in R2.

157 7.4. Further remarks

158 Chapter 8

Universal rigidity

Contents 8.1 Introduction ...... 160 8.2 Universal rigidity in R1 ...... 161 8.2.1 Universal rigidity of bipartite frameworks ...... 161 8.2.2 Bipartite graphs in general dimension ...... 164 8.2.3 Observations, open questions, conjectures ...... 165 8.3 Universal rigidity of tensegrity frameworks ...... 172 8.3.1 Introduction ...... 172 8.3.2 Preliminaries ...... 175 8.3.3 The conﬁguration matrix and Gale matrices ...... 177 8.3.4 Dominated tensegrity frameworks ...... 178 8.3.5 Aﬃnely dominated tensegrity frameworks ...... 180 8.3.6 Proof of main results ...... 183 8.3.7 Counter-example and conjectures ...... 185

159 8.1. Introduction

8.1 Introduction

First let us recall that a framework (G, p) in Rr is said to be in general position if for every U ⊆ V (G) with |U| ≥ r+1, the set {p(v): v ∈ U} is affinely independent. In dimension one, this condition simply means that no two vertices are mapped to the same point on the line. The affine dimension of a framework (G, p) is the dimension of the affine space spanned by {p(v): v ∈ V (G)}. A framework (G, p) in Rr is said to be (algebraically) generic if the set of all coordinates of p is algebraically independent over the rationals. Although the first result on universal rigidity of frameworks appeared in the work of Connelly [21] more than 30 years ago, the study of universal rigidity has attracted much interest only since the last decade, when algorithms using semidefinite programming are employed to solve network localization problem [15, 95]. Even when the solution is unique in 2 or 3-dimensional space, there is no guarantee that these algorithms return this solution. On the contrary, they tend to give a solution in a higher dimension. Hence universal rigidity is a desirable property to ensure the efficiency of these algorithms. However, it is likely that universal rigidity is more difficult to characterize than local rigidity and global rigidity. Still little is known about universal rigidity, even for generic frameworks in dimension 1, where the characterizations of local and global rigidity are almost trivial. Moreover, the fact that universal rigidity is not a generic property makes it difficult to handle. In this chapter, we study universal rigidity in two directions. The first one is to explore universal rigidity of bar-joint frameworks on the line. In contrast to local rigidity and global rigidity, the universal rigidity of frameworks seems quite complicated even in R1. Therefore, we start by investigating a special class of frameworks: complete bipartite frameworks. We obtain a complete characterization of the universal rigidity of these frameworks in general position on the line and deduce that the only bipartite graph that is generically universally rigid in Rd is

K1,1 for every d ≥ 1. This result suggests the study of other classes of frameworks which share some properties with bipartite frameworks as well as inspires several open questions and conjectures. The second direction is to relax the condition on the genericity of frameworks. As the study of universal rigidity arose from the quest for an eﬃcient algorithm for network localization problem, the requirement on genericity is far from sat- isfactory. An earlier work of Alfakih and Ye [8] gave suﬃcient condition for the universal rigidity of bar-joint frameworks in general position, a weaker require-

160 Chapter 8. Universal rigidity ment than genericity. We strengthen the results of Alfakih and Ye for bar-joint frameworks, allowing conﬁgurations in non general position. In fact, our result is obtained for tensegrity frameworks, a generalized model of bar-joint frameworks, where beside bars we have also cables, which prevent the distance between vertices from increasing, and struts, which prevent the distance between vertices from decreasing.

8.2 Universal rigidity in R1

The results in this section are from a joint work with Tibor Jord´an [70].

8.2.1 Universal rigidity of bipartite frameworks

Our main result is the following.

Theorem 8.2.1. Let G be a complete bipartite graph on at least three vertices with bipartition X = {x1, . . . , xm}, Y = {y1, . . . , yn} and p a realization of G in general position on the line.

1. If p(x1) < ··· < p(xm) < p(y1) < ··· < p(yn), then (G, p) is not universally rigid.

2. If p(x1) < ··· < p(xk) < p(y1) < ··· < p(yn) < p(xk+1) < ··· < p(xm) (or

symmetrically p(y1) < ··· < p(yk) < p(x1) < ··· < p(xm) < p(yk+1) < ··· <

p(yn)), then (G, p) is not universally rigid. 3. If none of the conditions above holds, then (G, p) is universally rigid.

In order to prove this theorem, we will need the following result.

Theorem 8.2.2 (Alfakih [4]). Let (G, p) be a framework in Rd. Then (G, p) has a non-zero positive semideﬁnite (PSD) stress matrix Ω if and only if (G, p) has no equivalent realization of aﬃne dimension |V | − 1.

We will also need some more deﬁnitions. Let (G, p) be a framework on the line with G = (V,E). A pair of vertices u, v ∈ V is called universally linked in (G, p) if kq(u) − q(v)k = kp(u) − p(v)k holds for all frameworks (G, q) which are equivalent to (G, p) (in all dimensions). Let C be a cycle of G passing through v1, . . . , vk with E(C) = {v1v2, . . . , vk−1vk, vkv1}. If p(v1) < p(v2) < ··· < p(vk) then C is called a stretched cycle in (G, p) (Figure 8.1). If C is a stretched cycle in (G, p)

161 8.2. Universal rigidity in R1 then it is not diﬃcult to see that every pair of vertices of C is universally linked in (G, p). In fact, the condition that the distance v1vk is the sum of the distance v1v2, . . . , vk−1vk implies that v1, . . . , vk must lie on a line and respect the order and distance between any pairs.

• • • • v1 v2 v3 v4

Figure 8.1: A stretched cycle on 4 vertices. The bar v1v4 is bent a little so that the other bars can be seen.

Now we are ready to prove Theorem 8.2.1.

Proof of Theorem 8.2.1.

1. Suppose that p(x1) < ··· < p(xm) < p(y1) < ··· < p(yn) holds and consider a PSD stress matrix Ω of (G, p). We will prove that Ω is the zero matrix.

Let rij = p(yj) − p(xi) > 0 denote the distance between xi and yj in (G, p), and wij the stress on the edge xiyj, for 1 ≤ i ≤ m and 1 ≤ j ≤ n. The equilibrium condition at vertices in X gives

rijwij = 0, for every i = 1, . . . , m. (8.2.1) j X

Let sj = p(yj) − p(y1) be the distance between y1 and yj. Then we have rij − ri1 = sj, for every i = 1, . . . , m and j = 1, . . . , n (see Figure 8.2.)

rij • ···• ···• ◦ · · · ◦ · · · ◦ x1 xi ri1 xm y1 sj yj yn

Figure 8.2: rij − ri1 = sj

n m The entries on the diagonal of Ω are j=1 wij, for i = 1, . . . , m, and i=1 wij, for j = 1, . . . , n. Since Ω is PSD, these entries are all non-negative. Therefore, P P n

ri1wij ≥ 0, for i = 1, . . . , m. j=1 X Using (8.2.1), we have

0 ≤ ri1wij = ri1wij − rijwij = (ri1 − rij)wij = − sjwij. j j j j j>1 X X X X X 162 Chapter 8. Universal rigidity

Therefore, j>1 sjwij ≤ 0, for j = 1, . . . , n. Then, since sj > 0 for j = 2, . . . , n,

P 0 ≤ sj wij = sjwij ≤ 0 j>1 i i j>1 X X X X which implies that equality holds everywhere. Thus, all entries on the diagonal of Ω are 0’s with possibly an exception of the entry corresponding to (x1, x1). However, by using the symmetry of the graph, we can deduce that this entry must also be 0. Therefore, the sum of all eigenvalues of Ω is 0. Hence Ω is the zero matrix. Theorem 8.2.2 now implies that (G, p) is not universally rigid, in fact, it has an equivalent realization of aﬃne dimension m + n − 1.

2. Suppose that p(x1) < ··· < p(xk) < p(y1) < ··· < p(yn) < p(xk+1) < ··· < p(xm) holds and consider a PSD stress matrix Ω of (G, p). We will prove that Ω is the zero matrix.

Let rij be the distance between xi and yj in this realization and wij the stress on the edge xiyj. Let

rij − ri1, for i ≤ k qj = ( ri1 − rij, for i ≥ k + 1 and rk+1,j + rij, for i ≤ k ti = ( rij − rk+1,j, for i ≥ k + 1

Then qj ≥ 0, ti ≥ 0 for every i, j and qj > 0 if j =6 1 and ti > 0 if i =6 k + 1.

Let Ai = ri1 j wij. Since j rijwij = 0 for every i by the equilibrium condition at vertices in X, we have P P

Ai = ri1 wij − rijwij j j X X = (ri1 − rij)wij j X − q w , for i ≤ k = j j ij q w , for i ≥ k + 1 ( Pj j ij

Let Bj = rk+1,j i wij. SinceP i≤k rijwij − i≥k+1 rijwij = 0 for every j = , . . . , n Y 1 by the equilibriumP conditionP at vertices inP , we have

Bj = (rk+1,j + rij)wij + (rk+1,j − rij)wij i k i k X≤ ≥X+1 = tiwij − tiwij. i k i k X≤ ≥X+1 163 8.2. Universal rigidity in R1

Therefore,

tiAi = tiAi + tiAi i i k i k X X≤ ≥X+1 = ti(− qjwij) + ti( qjwij) i k j i k j X≤ X ≥X+1 X = − qj tiwij + qj tiwij j i k j i k X X≤ X ≥X+1 = − qjBj. j X Since Ω is PSD and rij > 0, Ai,Bj ≥ 0 hold for every i, j. Hence 0 ≤ i tiAi = − q B ≤ 0 holds. Therefore, equality must occur everywhere, which means j j j P that A = 0 for i =6 k +1 and B = 0 for j =6 1. Thus every entry on the diagonal of P i j Ω with possible exceptions of the entries corresponding to (xk+1, xk+1) and (y1, y1) must be zero. However, by using the symmetry of the graph, we can deduce that these entries must also be zero, so every entry on the diagonal of the PSD matrix Ω is zero. Therefore, Ω is the zero matrix. Theorem 8.2.2 now implies that (G, p) is not universally rigid; in fact, it has an equivalent realization of aﬃne dimension m + n − 1.

3. If none of the conditions in 1 and 2 holds, then there exist, say x1, x2, y1, y2, such that p(x1) < p(y1) < p(x2) < p(y2). Then x1, y1, x2, y2 form a stretched cycle in

(G, p) and hence x1, x2 and y1, y2 are universally linked in (G, p). This implies that the pairwise distances among these four vertices are the same in all realizations of G equivalent to (G, p) and hence (G, p) is universally rigid if and only if (G′, p) is ′ ′ universally rigid, where G = G+x1x2 +y1y2. It remains to observe that (G , p) can be obtained from a framework on a complete graph on four vertices by iteratively attaching vertices of degree two (and adding edges). These operations are known to preserve universal rigidity on the line (Lemma 3.3.10). Therefore (G′, p) and hence (G, p) are universally rigid, as required.

Theorem 8.2.1 implies the following observation of Connelly. Corollary 8.2.3 (Connelly [30]). The only generically universally rigid bipartite 1 graph in R is the single edge K1,1.

8.2.2 Bipartite graphs in general dimension

The proof of Theorem 8.2.1 implies the following result in general dimension.

164 Chapter 8. Universal rigidity

Theorem 8.2.4. The only generically universally rigid bipartite graph in Rd is the single edge K1,1, for every dimension d ≥ 1. Proof. First we observe that no general position d-dimensional realization of a non-complete graph on at most d + 1 vertices is locally rigid. Thus the only d- dimensional generically universally rigid bipartite graph on at most d + 1 vertices is K1,1. Next consider a complete bipartite graph G on at least d + 2 vertices and a d-dimensional generic realization (G, p) with the property that the projected one-dimensional framework (G, p′), obtained by projecting the conﬁguration p to one coordinate axis, satisﬁes Condition 1 or 2 in Theorem 8.2.1 (Figure 8.3). ◦ • ◦ (K3,3, p) • ◦ •

′ (K3,3, p ) • • • ◦ ◦ ◦

Figure 8.3: A generic complete bipartite framework in R2 and its projection on a line.

A PSD stress matrix of (G, p) is also a PSD stress matrix of (G, p′) and hence, by the proof of Theorem 8.2.1, it must be the zero matrix. Thus, by Theorem 8.2.2,(G, p) is not universally rigid. Therefore, a complete bipartite graph, except the single edge K1,1, is not generically universally rigid in any dimension.

This result contrasts universal rigidity with local and global rigidity. While local and global rigidity are known to be implied by a high connectivity in dimension 1, 2 (and conjectured to be so in higher dimensions), Theorem 8.2.4 shows that high connectivity does not imply universal rigidity.

8.2.3 Observations, open questions, conjectures

The study of the universal rigidity of bipartite frameworks gives rise to many open questions and conjectures. In the remain of this section, we introduce and discuss these questions and conjectures, whose answers would provide deeper understanding of universal rigidity and could lead to combinatorial characterizations of universal rigidity at least in low dimensions. The ﬁrst question is motivated by Theorem 8.2.2.

165 8.2. Universal rigidity in R1

Question 1. Is it true that a generic framework (G, p) has a PSD stress matrix Ω of rank at least k if and only if (G, p) has no equivalent realization of aﬃne dimension |V | − i for any i where 1 ≤ i ≤ k.

Note that when k = 1 we have Theorem 8.2.2. The answer is “no” if we replace generic position by general position. (See the example in Section 8.3 Figure 8.10(b).) The “only if” part follows from the following result.

Theorem 8.2.5 (Alfakih [4]). Let (G, p) be a framework and Ω a PSD stress matrix of (G, p). Then Ω is a stress matrix for every framework (G, q) equivalent to (G, p).

(cf. Theorem 8.3.4) In fact, suppose that Ω is a PSD stress matrix of (G, p) of rank at least k, and (G, q) a framework equivalent to (G, p). Then Ω is a stress matrix for (G, q). If d is the aﬃne dimension of (G, q) then rank Ω ≤ |V |−d−1. Therefore, d ≤ |V |−k −1.

Theorem 8.2.1 also leads us to the following question.

Question 2. Is it true that the universal rigidity of a general position framework (G, p) in R1 depends only on the ordering of vertices on the line (and not on the coordinates)?

As remarked in the previous section, a highly connected complete bipartite graph is not generically universally rigid in R1 (1-GUR). However, if we add one edge to this graph, it becomes 1-GUR and many edges are redundant (Figure 8.4). This observation inspires the following question on minimally 1-GUR graphs, i.e., 1-GUR graph that is no more 1-GUR after the deletion of any edge.

•••···•

◦ ◦ ◦ · · · ◦

Figure 8.4: A sparse 1-GUR subgraph of a complete bipartite graph plus one edge.

Question 3. Let G = (V,E) be a minimally 1-GUR graph. Does there exists a (linear) upper bound on |E| as a function of |V |?

166 Chapter 8. Universal rigidity

Now let us recall the Ratmanski’s construction in Lemma 3.3.10: if we glue two 1-GUR graphs over at least 2 vertices then we obtain another 1-GUR graph. In order to prove the conjecture that the converse is also true, one should study the eﬀect of cleavage on a 1-GUR graph. Let G = (V,E) be a graph. A pair (G1,G2), where G1,G2 are subgraphs of G, is called a k-separator of G if V (G1)∪V (G2) = V ,

E(G1) ∪ E(G2) = E, and |V (G1) ∩ V (G2)| = k hold. For a subset X ⊆ V let G + K(X) denote the supergraph of G obtained by adding all edges connecting pairs of vertices of X (which are non-adjacent in G). We have the following simple result.

Lemma 8.2.6. Let G be a 1-GUR graph and (G1,G2) a k-separator of G with

X = V (G1) ∩ V (G2). Then Gi + K(X) is 1-GUR for i = 1, 2.

Proof. Suppose that Gˆ1 = G1 + K(X) is not 1-GUR. Then there exists a generic 1 realization (Gˆ1, p1) of Gˆ1 in R which is not univerally rigid and hence there exists ˆ ′ ˆ a realization (G1, p1) equivalent but non congruent to (G1, p1). We can assume ′ that p1(v) = p1(v) for every v in X. Extend p1 to a generic realization p of G in R1. Let p′ (v), v ∈ V (G ) p′(v) = 1 1 ( p(v), v ∈ V (G2) Then (G, p′) is equivalent but not congruent to (G, p), which means that G is not 1-GUR, a contradiction.

In particular, a 1-GUR graph can be cut into two 1-GUR graphs along a cut pair u, v if we add the edge uv into both sides. The addition of this edge uv is undesirable when uv is not originally presented in G. What one may expect is that one of the two graphs in the 2-separator is 1-GUR.

Conjecture 8.2.7. Let G be 1-GUR and (G1,G2) a 2-separation in G with V (G1)∩

V (G2) = {x, y}. Then G1 or G2 is 1-GUR.

The following conjecture might be helpful in proving this. A pair of vertices {u, v} in G is called universally linked in Rd if {u, v} is universally linked in all d-dimensional generic realizations (G, p) of G.

Conjecture 8.2.8. Suppose that a pair {u, v} is not universally linked in G in R1. Then there exist generic 1-dimensional realizations (G, p), (G, q) of G and a realization (G, p′) equivalent to (G, p), a realization (G, q′) equivalent to (G, q), such that kp′(u) − p′(v)k > kp(u) − p(v)k and kq′(u) − q′(v)k < kq(u) − q(v)k.

167 8.2. Universal rigidity in R1

By assuming the truth of Conjecture 8.2.8 we can show that if G1 and G2 are not 1-GUR then there exists a general position realization (G, p) on the line which is not universally rigid. We believe that p can be made generic, too (c.f. Question 2), which would imply Conjecture 8.2.7.

Our argument is as follows. We may assume that {x, y} is not universally 1 linked in G1 and G2. Thus there is a generic realization (G1, p) in R and an equivalent realization (G1, q) such that the distance between p(x) and p(y) is, say, stricly smaller than the distance between q(x) and q(y). By assuming the truth of ′ 1 Conjecture 8.2.8 we can ﬁnd a generic realization (G2, p ) in R and an equivalent ′ ′ ′ realization (G2, q ) such that the distance between p (x) and p (y) is, say, stricly smaller than the distance between q′(x) and q′(y). By carefully choosing the generic ′ realization (G2, p ) and rescaling, if necessary, we may assume that kp(x)−p(y)k = ′ ′ kp (x)−p (y)k. Now using Lemma 3.2.5 we can obtain a pair of realizations (G1, r) ′ ′ ′ and (G2, r ) for which kr(x)−r(y)k = kr (x)−r (y)k > kp(x)−p(y)k and such that ′ ′ (G1, r) is equivalent to (G1, p) and (G2, r ) is equivalent to (G2, p ). By glueing ′ ′ together (G1, p) and (G2, p ) as well as (G1, r) and (G2, r ) along the pair x, y we obtain two equivalent but not congruent realizations of G, where the former realization is 1-dimensional. The conﬁguration obtained by glueing (G1, p) and ′ (G2, p ) together may not be generic. However, we believe that a generic non 1-UR framework can be obtained if we perturb it to a close enough generic one on the line.

Another possible approach to the diﬃcult problem of characterizing universal rigidity is study classes of non generically universally rigid (non GUR) graphs. The proof of Theorem 8.2.1 shows that a bipartite framework on the line is not universally rigid when there is no stretched cycle. Therefore, the class of graphs which have a realization on the line without stretched cycles is an interesting candidate for non 1-GUR graphs. In fact, this class of graphs is known as cover graphs and they can be deﬁned in the following manner. Let G = (V,E) be a graph and let G~ be an acyclic orientation of G. An edge e of G is dependent if the reversal of e in G~ creates a directed cycle. An orientation without dependent edges is called strongly acyclic. We say that G is a cover graph if G has a strongly acyclic orientation. (It is known that G is a cover graph if and only if it is the Hasse diagram of some partially ordered set on V .) One can see that a cover graph can be realized as a framework on the line without stretched cycles: we can order the vertices of a cover graph on the line so that if uv is an arc then u is on the right of v. Conversely, a realization as a framework on the line without stretched

168 Chapter 8. Universal rigidity cycles of a graph naturally gives a strongly acyclic orientation of the graph by orienting every edges from left to right. Furthermore, cover graphs are triangle-free. Ratmanski’s construction adds more reason to think that the following question has an aﬃrmative answer.

Question 4. Is it true that no cover graph is 1-GUR (except K1,1)?

We should also remark that it is NP-hard to test whether a given graph is a cover graph [17, 84]. Some special subclasses of cover graphs may also be interesting. One of them is triangle-free 3-colorable graphs, for which a realization on the line without stretched cycles looks quite similar to the second conﬁguration in Theorem 8.2.1. These graphs are known to be cover graphs, a fact followed from a more general result that every k-colorable graph without cycles of length at most k is a cover graph [34]. Also, one may ﬁrst check the question for triangle-free planar graphs.

Question 5. Is it true that no triangle-free planar graph (or even triangle-free

3-colorable graph) is 1-GUR (except K1,1)?

We may also ask whether all non cover graphs are 1-GUR. An interesting graph to analyze is the Grötzsch graph, which is triangle-free and 4-chromatic, see Figure 8.5. This graph is not a cover graph [34]. Is it 1-GUR? Since it is triangle-free, an affirmative answer to this question will imply that Ratmanski’s construction is not sufficient to generate all 1-GUR graphs.

Figure 8.5: The Gr¨otzsch graph.

Sparse graphs and cover graphs

The rest of this section contains some further questions and observations about the relation between sparse graphs and cover graphs. Let G = (V,E) be a simple graph. We recall that G is (2, 4)-sparse if for all subsets X ⊆ V with |X| ≥ 3 the

169 8.2. Universal rigidity in R1 subgraph induced by X has at most 2|X| − 4 edges. For example, triangle-free planar graphs are (2, 4)-sparse.

Question 6. Is every (2, 4)-sparse graph a cover graph?

A (2, 4)-sparse graph is clearly triangle-free. It is also independent in the 2- dimensional generic rigidity matroid by Laman’s theorem. This leads us to a further extension:

Question 7. Is every triangle-free graph which is independent in the 2-dimensional generic rigidity matroid a cover graph?

One proof method for a positive result here would use the Henneberg operations. The next lemmas on the construction of cover graphs show that this approach may be useful.

Lemma 8.2.9. Let G be a triangle-free graph obtained from a graph H by a 0- extension operation. Then G is a cover graph if and only if H is a cover graph.

Proof. Since H is a subgraph of G, necessity is obvious. To see the other direction consider a strongly acyclic orientation H of H. Suppose that G = H + vx + vy. Since H is acyclic, we cannot have an (x, y)-directed path and a (y, x)-directed path in H simultaneously. Thus we have three cases to consider. Case 1: There is an (x, y)-directed path in H. Then we orient vx from x to v and vy from v to y. Case 2: There is an (y, x)-directed path in H. Then we orient vx from v to x and vy from y to v. Case 3: There is neither (x, y)-directed path nor (y, x)-directed path in H. Then we orient vx from v to x and vy from v to y. Lemma 8.2.10. Let G be a triangle-free graph obtained from a cover graph H by a 1-extension operation. Then G is also a cover graph.

Proof. Consider a strongly acyclic orientation H of H. Suppose that G = H − xy + vx + vy + vz. We orient the edges vx, vy, vz as follows. Case 1: There is an (x, z)-directed path in H −xy. Then there is no (z, y)-directed path in H − xy. Case 1.1: There is a (y, z)-directed path in H − xy. We orient vx from x to v, vy from y to v and vz from v to z. (Figure 8.6)

170 Chapter 8. Universal rigidity

Figure 8.6: Cases 1.1 and 1.2

Case 1.2: There is no (y, z)-directed path in H − xy. We orient vx from x to v, vy from v to y and vz from v to z. (Figure 8.6)

Case 2: There is an (z, x)-directed path in H − xy. Then every (y, z)-path in H − xy has at least two backward edges. We orient vx from v to x, vy from v to y and vz from z to v. (Figure 8.7)

Figure 8.7: Case 2 and Case 3.1

Case 3: There is neither (x, z)-directed path nor (z, x)-directed path in H − xy. Case 3.1: There is a (y, z)-directed path in H − xy. Then every (x, z)-path in H − xy has at least two forward edges. We orient vx from x to v, vy from y to v and vz from v to z. (Figure 8.7) Case 3.2: There is a (z, y)-directed path in H − xy. We orient vx from x to v, vy from v to y and vz from z to v. (Figure 8.8)

Figure 8.8: Cases 3.2 and 3.3

Case 3.3: There is neither (y, z)-directed path nor (z, y)-directed path in H − xy. We orient vx from x to v, vy from y to v and vz from z to v. (Figure 8.8)

171 8.3. Universal rigidity of tensegrity frameworks

8.3 Universal rigidity of tensegrity frameworks

8.3.1 Introduction

In this section, we make a convention that

all vectors (or points) are considered to be column vectors.

An r-dimensional tensegrity framework is a pair (G, p) where G = (V,E) is a graph whose edges are labelled as a bar, a cable or a strut and p is an assignment of a point p(v) in Rr to each vertex v ∈ V . For ease of notation, in this section, we will identify the vertex set V with {1, . . . , n} where n = |V | and write pi instead of p(i). The affine dimension of the configuration p or the framework (G, p), is the dimension of the affine space spanned by p1, . . . , pn. If the affine dimension of p is equal to the embedding dimension r then we say that (G, p) is of full dimension. Let B, C and S denote the sets of bars, cables and struts of (G, p) respectively. Thus E = B ∪ C ∪ S. A bar framework (which is in fact our familiar bar-joint framework) (G, p) is a tensegrity framework where E = B, i.e., C = S = ∅. Let (G, p) and (G, q) be two tensegrity frameworks in Rr and Rs respectively. (G, q) is said to be congruent to (G, p) if

kqi − qjk2 = kpi − pjk2 for all i, j = 1, . . . , n.

On the other hand, we say that (G, q) is dominated by (G, p), (or (G, p) dominates (G, q)), if kqi − qjk2 = kpi − pjk2 for each {i, j} ∈ B, kqi − qjk2 ≤ kpi − pjk2 for each {i, j} ∈ C, (8.3.1) kqi − qjk2 ≥ kpi − pjk2 for each {i, j} ∈ S. A tensegrity framework (G, q) in Rr is said to be aﬃnely-dominated by (G, p) if (G, q) is dominated by (G, p) and qi = Api + b for all i = 1, . . . , n, where A is an r × r matrix and b ∈ Rr. A tensegrity framework (G, p) in Rr is said to be dimensionally rigid if no tensegrity framework (G, q) in any dimension ≥ r + 1 is dominated by (G, p). We say that a tensegrity framework (G, p) is universally rigid if every tensegrity framework (G, q) in any dimension that is dominated by (G, p) is in fact congruent to (G, p).

172 Chapter 8. Universal rigidity

Let (G, p) be a tensegrity framework on n vertices in Rr. An equilibrium stress of (G, p) is a real valued function ω on E such that

i j ωij(p − p ) = 0 for all i = 1, . . . , n. (8.3.2) j:{Xi,j}∈E

An equilibrium stress ω is said to be proper if ωij ≥ 0 for all {i, j} ∈ C and

ωij ≤ 0 for all {i, j} ∈ S. Let E denote the set of missing edges in G, i.e.,

E = {{i, j} : i =6 j, {i, j} 6∈ E}, and let ω be an equilibrium stress of (G, p). Then the n × n symmetric matrix Ω where −ωij if {i, j} ∈ E, 0 if {i, j} ∈ E, Ωij =  (8.3.3)  ω if i = j,  ik k:{Xi,k}∈E  is called the stress matrix associated with ω, or a stress matrix of (G, p). The stress matrix Ω associated with a proper equilibrium ω is called proper. The following result provides a suﬃcient condition for the universal rigidity of a given tensegrity framework.

Theorem 8.3.1 (see [76, 21, 23]). Let (G, p) be a tensegrity framework of full dimension in Rr, for some r ≤ n − 2. Suppose that the following two conditions hold.

1. There exists a proper PSD stress matrix Ω of (G, p) of rank n − r − 1.

2. There does not exist a tensegrity framework (G, q) in Rr that is aﬃnely- dominated by, but not congruent to, (G, p).

Then (G, p) is universally rigid.

Note that (n − r − 1) is the maximum possible value for the rank of the stress matrix Ω. As we will see in the next, the existence of a proper PSD stress matrix of maximal rank in condition 1 implies that every framework dominated by (G, p) is obtained from (G, p) by an aﬃne transformation. Condition 2 excludes those frameworks that are not congruent to (G, p). When (G, p) is a generic framework, condition 2 in Theorem 8.3.1 is implied by condition 1.

173 8.3. Universal rigidity of tensegrity frameworks

Theorem 8.3.2 ([21, 23, 76]). Let (G, p) be a generic tensegrity framework of full dimension in Rr for some r ≤ n − 2. If (G, p) has a proper PSD stress matrix Ω with rank Ω = n − r − 1 then (G, p) is universally rigid.

Alfakih and Ye showed that the condition on genericity in the previous theorem can be weakened for bar frameworks. A conﬁguration p is said to be in general position if every subset of {p1, . . . , pn} of cardinality at most r + 1 is aﬃnely independent.

Theorem 8.3.3 (Alfakih and Ye [8]). Let (G, p) be a bar framework on n vertices in general position in Rr with r ≤ n − 2. If (G, p) has a proper PSD stress matrix Ω with rank Ω = n − r − 1 then (G, p) is universally rigid.

Contribution: In this section we ﬁrst extend these suﬃcient conditions to tensegrity frameworks in general position.

Theorem 8.3.4. Let (G, p) be a tensegrity framework on n vertices in general position in Rr for some r ≤ n − 2. Suppose that (G, p) is of full dimension. If (G, p) has a PSD stress matrix Ω with rank Ω = n−r −1 then (G, p) is universally rigid.

For bar frameworks, the result can be furthermore strengthened. For a vertex i of {1, . . . , n}, let N(i) denote the set of neighbors of i, that is, the set of vertices adjacent to i.

Theorem 8.3.5. Let (G, p) be a bar framework on n vertices in Rr for some r ≤ n − 2. Suppose that the following conditions hold.

1. (G, p) has a PSD stress matrix Ω with rank Ω = n − r − 1 2. For each vertex i ∈ {1, . . . , n}, the set {pj : j ∈ {i} ∪ N(i)} aﬃnely spans Rr.

Then (G, p) is universally rigid.

The proofs of these results will be given in Section 8.3.6. To this end we characterize dominated tensegrity frameworks and aﬃnely dominated tensegrity frameworks in Section 8.3.4 and 8.3.5 respectively. Lastly, Section 8.3.7 discusses an example showing that the converse of Theorem 8.3.4 is not true. The results in this section are from a joint work with Abdo Alfakih [8].

174 Chapter 8. Universal rigidity

• •

3 −3 −1 −1 1 1 • • 3 3 −3 −3 •• •• −1 1

(a) (b)

Figure 8.9: Two tensegrity frameworks. Cables are denoted by dashed lines and struts by double lines. (a) The framework is universally rigid. The associated proper stress matrix is PSD of rank 1=4 − 2 − 1. (b) The framework is not universally rigid. The only proper stress matrix (up to positive multiple) is negative semideﬁnite.

Example: Consider two tensegrity frameworks in the plane in Figure 8.9 where the outer triangles are regular and the fourth vertices are in the center. We denote cables by dashed lines and struts by double lines. The unique proper stress (up to a positive scalar multiple) is indicated on the ﬁgure in each case. It is easy to see that the framework in (a) is universally rigid while the framework in (b) is not. The corresponding proper stress matrices are

1 1 1 −3 1 1 1 −3 Ω1 =   and Ω2 = −Ω1. 1 1 1 −3   −3 −3 −3 9     

Ω1 is PSD and rank Ω1 = 1 = 4 − 2 − 1 while Ω2 is negative semideﬁnite.

8.3.2 Preliminaries

Let ei denote the ith unit vector in Rn and let 1 denote the vector of all 1’s in Rn. Deﬁne the following n × n symmetric matrices where i < j.

F ij = (ei − ej)(ei − ej)T , Eij = ei(ej)T + ej(ei)T , (8.3.4) Li = ei1T + 1(ei)T .

175 8.3. Universal rigidity of tensegrity frameworks

Recall the Kronecker delta δij deﬁned by

1 if i = j, δij = ( 0 otherwise.

Then the following two lemmas easily follow.

Lemma 8.3.6. We have the following equalities.

kl ij hF ,E i = −2δkiδlj (8.3.5) for 1 ≤ k < l ≤ n, 1 ≤ i < j ≤ n, and

hF kl ,Lii = 0, (8.3.6) for 1 ≤ k < l ≤ n and 1 ≤ i ≤ n.

Equation (8.3.5) implies that F kl is orthogonal to Eij if (k, l) =6 (i, j) and equation (8.3.6) implies that F kl is orthogonal to Li for every 1 ≤ k < l ≤ n and 1 ≤ i ≤ n.

Lemma 8.3.7.

1. The set {F ij : 1 ≤ i < j ≤ n} is linearly independent.

2. The set {Eij : 1 ≤ i < j ≤ n} ∪ {Li : 1 ≤ i ≤ n} is linearly independent.

An immediate consequence of Lemmas 8.3.6 and 8.3.7 is as follows.

Corollary 8.3.8. Let N be a subset of {(i, j) : 1 ≤ i < j ≤ n} and L the linear space spanned by {F ij :(i, j) ∈ N}. Then the linear space spanned by {Eij : 1 ≤ i < j ≤ n, (i, j) ∈/ N} ∪ {Li : 1 ≤ i ≤ n} is the orthogonal space L⊥ of

L in Sn.

Proof. By Lemma 8.3.6, the linear subspace spanned by {Eij : 1 ≤ i < j ≤ n, (i, j) ∈/ N} ∪ {Li : 1 ≤ i ≤ n} is orthogonal to the linear space L spanned by {F ij :(i, j) ∈ N}. Moreover, by Lemma 8.3.7, {F ij :(i, j) ∈ N} is linearly independent hence a base of L. Also, {Eij : 1 ≤ i < j ≤ n} ∪ {Li : 1 ≤ i ≤ n} is linearly independent. A simple count shows that the co-dimension of L is equal to the cardinality of {Eij : 1 ≤ i < j ≤ n} ∪ {Li : 1 ≤ i ≤ n}. The result follows.

176 Chapter 8. Universal rigidity

8.3.3 The conﬁguration matrix and Gale matrices

The conﬁguration matrix P of p is the n × r matrix whose i-th row is (pi)T , i.e.,

(p1)T . P =  .  (pn)T     Then we have hF ij ,PP T i = kpi − pjk2 for 1 ≤ i < j ≤ n, (8.3.7) since hF ij ,PP T i = trace (F ijPP T ) = trace ((ei − ej)(ei − ej)T PP T ) = (ei − ej)T PP T (ei − ej) = (pi − pj)(pi − pj)T = kpi − pjk2. The augmented conﬁguration matrix Pˆ is the n×(r+1) matrix obtained from P by adding an all-one column vector to the right of P . Then an embedding p in Rn is in general position if and only if every r + 1 rows of Pˆ are linearly independent. It is easy to see that if Ω is a stress matrix for (G, p) then ΩPˆ = 0. Therefore we have the following simple but important consequence. Lemma 8.3.9. If Ω is a stress matrix for a framework (G, p) then the aﬃne dimension of p is at most n − 1 − rank Ω.

Proof. The affine dimension of p is affdim(p) = rank Pˆ −1. Since ΩPˆ = 0 we have rank Ω ≤ n−rank Pˆ ≤ n−affdim(p)−1. Therefore affdim(p) ≤ n−1−rank Ω.

Suppose that p is a configuration of {1, . . . , n} in Rr that affinely spans Rr. Letr ¯ denote n − r − 1. A Gale matrix of p is an n × r¯ matrix Z whose columns form a base of the nullspace of PˆT . Let (zi)T be the i-th row of Z. The vector zi is called the Gale transformation of pi, for i = 1, . . . , n. The following lemma is easy but useful. Lemma 8.3.10. If Z is a Gale matrix of a configuration p and Q is any non- singular r¯×r¯ matrix then Z′ = ZQ is also a Gale matrix of p. On the other hand, if Z,Z′ are Gale matrices of p then there exists a non-singular r¯×r¯ matrix Q such that Z′ = ZQ.

Proof. If Z is a Gale matrix of a conﬁguration p and Z′ = ZQ for a non-singular r¯×r¯ matrix Q then the columns of Z′ are linearly independent. Moreover, PˆT Z′ = PˆT ZQ = 0. Hence, Z′ is a Gale matrix of p. Conversely, if Z,Z′ are Gale matrices of p then the columns of Z and the columns of Z′ form bases of the same linear space. Therefore, Z′ = ZQ for some non-singularr ¯ × r¯ matrix Q.

177 8.3. Universal rigidity of tensegrity frameworks

For a matrix M and a subset J of the row index set of M, we denote by MJ the submatrix of M whose rows are indexed by J. The following observation on the relation between the augmented conﬁguration matrix and Gale matrix is crucial in the proof of our main theorems.

Lemma 8.3.11. Let (G, p) be a framework in Rr and Z a Gale matrix of p. ˆ Suppose that J ⊆ V satisﬁes |J| = r + 1 and PJ is non-singular. Then ZJ¯ is non-singular, where J¯ = V \ J.

¯ i Proof. Suppose that λi for i ∈ J are scalars satisfying i∈J¯ λiz = 0. We show that λ = 0 for all i ∈ J¯. i P T T Set λi = 0 for i ∈ J and λ = (λ1, . . . , λn) , we have Z λ = 0. Since the columns of Pˆ form a base of the nullspace of ZT , we deduce that λ = Pˆ x for some r+1 x ∈ R . Then, 0 = λJ = PˆJ x holds. The non-singularity of PˆJ implies that x = 0. Thus λ = 0. This completes the proof.

Corollary 8.3.12. Let (G, p) be a framework in Rr and Z a Gale matrix of p. Suppose that J ⊆ V satisﬁes that the set {pi : i ∈ J} aﬃnely spans Rr. Then the ¯ rows of the matrix ZJ¯ are linearly independent, where J = V \ J.

Proof. Let J ′ ⊆ J satisfy |J ′| = r + 1 and {pi : i ∈ J ′} is aﬃnely independent. ′ ′ Then Lemma 8.3.11 applied to J implies that ZJ¯′ is non-singular. Since J ⊆ J, the rows of ZJ¯ are also rows of ZJ¯′ so the corollary follows.

8.3.4 Dominated tensegrity frameworks

Without loss of generality we assume that

Assumption 8.3.1. For every conﬁguration p, the centroid of the points p1, . . . , pn coincides with the origin, i.e., P T 1 = 0.

For ease of notation, we deﬁne

ij E(y) = yijE , (8.3.8) {i,j}∈XE∪C∪S |E|+|C|+|S| where y = (yij) ∈ R . The next lemma characterizes dominated tensegrity frameworks in terms of the matrix PP T (which is often called the Gram matrix of the conﬁguration p).

178 Chapter 8. Universal rigidity

Lemma 8.3.13. Let (G, p) and (G, p′) be two tensegrity frameworks in Rr and Rs of the same labeled graph G. Then (G, p′) is dominated by (G, p) if and only if P ′P ′T − PP T = E(y) + x1T + 1xT , (8.3.9)

n for some y = (yij) and x = (xi) ∈ R where yij ≥ 0 for all {i, j} ∈ C and yij ≤ 0 for all {i, j} ∈ S.

Proof. Let L = span {F ij : {i, j} ∈ B}. Then it follows from Lemmas 8.3.6 and 8.3.7 that {Eij : {i, j} ∈ E ∪ C ∪ S} ∪ {Li : i = 1, . . . , n} is a basis for L⊥, the ′ orthogonal complement of L in Sn. Now, by equation (8.3.7), (G, p ) is dominated by (G, p) if and only if hF ij ,P ′P ′T − PP T i = 0 for all {i, j} ∈ B, (8.3.10) hF ij ,P ′P ′T − PP T i ≤ 0 for all {i, j} ∈ C, (8.3.11) hF ij ,P ′P ′T − PP T i ≥ 0 for all {i, j} ∈ S. (8.3.12) But (8.3.10) holds if and only if (P ′P ′T − PP T ) ∈ L, i.e., P ′P ′T − PP T = ij n i T T {i,j}∈E∪C∪S yijE + i=1 xiL = E(y)+xe +ex for some y and x. On the other hand, (8.3.11) holds if and only if y ≥ 0 for all {i, j} ∈ C since, for {k, l} ∈ C, it P P ij follows from Lemma 8.3.6 that

kl ′ ′T T kl ij hF ,P P − PP i = hF ,E iyij = −2ykl ≤ 0. {i,jX}∈C Similarly, (8.3.12) holds if and only if yij ≤ 0 for all {i, j} ∈ S.

The following result reveals a crucial property that a proper PSD stress matrix of a framework is a stress matrix for every dominated framework. Moreover, cables and struts with non-zero stresses behave as bars. Theorem 8.3.14. Let (G, p) be a given tensegrity framework and Ω a proper PSD stress matrix of (G, p). Suppose that (G, p′) is a tensegrity framework dominated by (G, p). Then the following holds. 1. Ω is a stress matrix for (G, p′).

′i ′j i j 2. For every {i, j} ∈ C ∪ S such that ωij =6 0 we have kp − p k = kp − p k. Proof. Let (G, p′) be a tensegrity framework dominated by (G, p). Then, by equation (8.3.7) and deﬁnition,

ij ′ ′T T ωij hF ,P P − PP i = 0 for each {i, j} ∈ B, ij ′ ′T T ωij hF ,P P − PP i ≤ 0 for each {i, j} ∈ C, ij ′ ′T T ωij hF ,P P − PP i ≤ 0 for each {i, j} ∈ S.

179 8.3. Universal rigidity of tensegrity frameworks

Therefore,

′ ′T T ij ′ ′T T hΩ ,P P − PP i = ωij hF ,P P − PP i ≤ 0. {i,j}∈XB∪C∪S But ΩPP T = 0. Therefore, hΩ ,P ′P ′T i ≤ 0. However, both P ′P ′T and Ω are positive semideﬁnite. Therefore, hΩ ,P ′P ′T i = 0 and hence ΩP ′P ′T = 0 by Lemma 2.5.3. It follows that Ω is a stress matrix of (G, p′). ij ′ ′T T The above argument also implies that ωij hF ,P P −PP i = 0 for all {i, j} ∈ ij ′ ′T T E. Hence, for {i, j} ∈ C ∪ S with wij =6 0 we have hF ,P P − PP i = 0, thus kp′i − p′jk = kpi − pjk.

Using Theorem 8.3.14, the following theorem easily follows. Theorem 8.3.15. Let (G, p) be a given tensegrity framework on n vertices in Rr with r ≤ n − 2 and let Ω be a proper PSD stress matrix of (G, p) of rank n − r − 1. Then (G, p) is dimensionally rigid.

Proof. Let (G, p′) be a tensegrity framework of full dimension in Rs dominated by (G, p). Then the dimension of the nullspace of Ω is r + 1. Moreover, it follows from Theorem 8.3.14 that ΩP ′P ′T = 0. On the other hand, 1T Ω = 0, thus 1 is not in the column space of Ω. Also we have 1T P ′ = 0 by Assumption 8.3.1. Therefore, rank P ′ ≤ r and hence, (G, p) is dimensionally rigid.

Proof of Theorem 8.3.1 Let (G, p′) be a tensegrity framework in Rs dominated by (G, p) and let P ′ be the conﬁguration matrix of (G, p′). Theorem 8.3.15 implies that we can assume that s = r. Indeed, since (G, p) is dimensionally rigid, if s > r we can transform (G, p′) to a congruent framework in Rr and if s < r we can consider (G, p′) as embedded in Rr. Also, by the proof of Theorem 8.3.15, we

′ Ω have that the columns of P belong to the nullspace of T . Then there exists 1 ! an r × r matrix A such that P ′ = PA since the columns of P form a basis of this null space. Hence, the conﬁguration p′ is obtained from p by an aﬃne transformation. Hence, (G, p′) is congruent to (G, p) and thus (G, p) is universally rigid. 2

8.3.5 Aﬃnely dominated tensegrity frameworks

By Theorem 8.3.1, the main task in proving Theorem 8.3.4 is to exclude aﬃnely- dominated but non congruent frameworks. The next result which is implicit in [76] characterizes when this happens.

180 Chapter 8. Universal rigidity

Lemma 8.3.16. Let (G, p) be a tensegrity framework in Rr. Then there exists a tensegrity framework (G, p′) aﬃnely-dominated by, but not congruent to, (G, p) if and only if there exists a non-zero symmetric r × r matrix Φ such that

hF ij ,P ΦP T i = 0 for all {i, j} ∈ B, (8.3.13) hF ij ,P ΦP T i ≤ 0 for all {i, j} ∈ C, (8.3.14) hF ij ,P ΦP T i ≥ 0 for all {i, j} ∈ S. (8.3.15)

Proof. To prove the “only if” part, assume that (G, p′) is aﬃnely-dominated by, but not congruent to, (G, p). Then P ′ = PA for some r × r matrix A. Thus, by (8.3.7), kp′i − p′jk2 − kpi − pjk2 = hF ij ,P (AAT − I)P T i.

The result follows by setting Φ = AAT − I. Obviously, Φ is symmetric. Further- more, A is not orthogonal since (G, p′) is not congruent to (G, p) thus Φ =6 0. To prove the “if” part assume that there exists a non-zero symmetric matrix Φ satisfying (8.3.13)-(8.3.15). Then there exists a suﬃciently small ǫ > 0 such that I + ǫΦ is positive deﬁnite. Thus there exists an r × r nonsingular matrix A such that AAT = I + Φ. Hence,

hF ij ,P ΦP T i = hF ij ,P (AAT − I)P T i = kp′i − p′jk2 − kpi − pjk2.

Thus the result follows.

If a proper stress matrix of (G, p) is known, then Lemma 8.3.16 can be strengthened. (Also implicit in [76].)

Lemma 8.3.17. Let (G, p) be a tensegrity framework in Rr and let Ω be a proper stress matrix of (G, p). Then there exists a tensegrity framework (G, p′) aﬃnely- dominated by, but non congruent to, (G, p) if and only if there exists a nonzero symmetric r × r matrix Φ such that:

hF ij ,P ΦP T i = 0 for all {i, j} ∈ B ∪ C∗ ∪ S∗, (8.3.16) hF ij ,P ΦP T i ≤ 0 for all {i, j} ∈ C0, (8.3.17) hF ij ,P ΦP T i ≥ 0 for all {i, j} ∈ S0, (8.3.18)

∗ ∗ 0 ∗ where C = {{i, j} ∈ C : ωij =6 0}, S = {{i, j} ∈ S : ωij =6 0}, C = C \ C and S0 = S \ S∗.

181 8.3. Universal rigidity of tensegrity frameworks

Proof. Let Φ be the matrix deﬁned as in Lemma 8.3.16. By deﬁnition, ΩP ΦP T = 0. Therefore,

T ij T hΩ ,P ΦP i = ωijhF ,P ΦP i {i,jX}∈E ij T = ωijhF ,P ΦP i {i,jX}∈C∪S = 0,

ij T ij T since hF ,P ΦP i = 0 for every {i, j} ∈ B. But ωijhF ,P ΦP i ≤ 0, for every ij T ij T {i, j} ∈ C, since ωij ≥ 0 and hF ,P ΦP i ≤ 0. Similarly, ωijhF ,P ΦP i ≤ 0, for every {i, j} ∈ S. Therefore,

ij T ωijhF ,P ΦP i = 0 for each {i, j} ∈ C ∪ S.

Thus, hF ij ,P ΦP T i = 0 for each {i, j} ∈ C∗ ∪ S∗ and the result follows.

The next result allows us to translate (8.3.16)-(8.3.18) into a condition on Gale matrices which is easier to deal with since a Gale matrix with nice property can be obtained from a stress matrix of maximal rank. Lemma 8.3.18. Let (G, p), Ω,C0,S0,C∗,S∗ be deﬁned as in Lemma 8.3.17 and Z be any Gale matrix of p. If there exists a tensegrity framework (G, p′) aﬃnely- dominated by, but non congruent to, (G, p) then there exist a non-zero y = (yij) ∈ |E|+|C0|+|S0| r¯ R and ξ = (ξi) ∈ R such that

E ∗(y)Z = 1ξT , (8.3.19)

∗ ij where E (y) = {i,j}∈E∪C0∪S0 yijE .

Proof. AssumeP that there exists a non-zero symmetric Φ satisfying (8.3.16)-(8.3.18). Let L = span {F ij : {i, j} ∈ B ∪ C∗ ∪ S∗}. Then it follows from Lemmas 8.3.6 and 8.3.7 that {Eij : {i, j} ∈ E ∪C0 ∪S0} ∪{Li : i = 1, . . . , n} is a basis for L⊥, the or- ij T ∗ ∗ thogonal complement of L in Sn. Since hF ,P ΦP i = 0 for all {i, j} ∈ B∪C ∪S if and only if P ΦP T ∈ L⊥, we have,

n T ij i ∗ T T P ΦP = yijE + xiL = E (y) + x1 + 1x 0 0 i=1 {i,j}∈XE∪C ∪S X for some y and x. Therefore, y and ξ = −ZT x is a solution of (8.3.19) since P ΦP T Z = 0 = E ∗(y)Z + 1xT Z.

182 Chapter 8. Universal rigidity

8.3.6 Proof of main results

We need some more lemmas. The ﬁrst one simpliﬁes equation (8.3.19) when Z has a nice property. Lemma 8.3.19. Let Z be a n × r¯ matrix whose rows are indexed by {1, . . . , n} and whose columns are indexed by {j1, . . . , jr¯} with the property

0 0 zijk = 0 for {i, jk} ∈ E ∪ C ∪ S .

Then the equation E ∗(y)Z = 1ξT is equivalent to E ∗(y)Z = 0.

Proof. For every k = 1,..., r¯,

n ∗ ξjk = E (y)jkizijk i=1 X ∗ ∗ ∗ = E (y)jkizijk + E (y)jkizijk + E (y)jkjk zjkjk i i,j E :{ Xk}∈ i:{i,jXk}∈E = 0

∗ ∗ ∗ 0 0 since if {i, jk} ∈ B ∪ C ∪ S then E (y)ijk = 0 while if {i, jk} ∈ E ∪ C ∪ S then ∗ zijk = 0. Therefore E (y)Z = 0 holds. The converse is trivial.

When the stress matrix Ω has maximal rank then a Gale matrix with the nice property as in Lemma 8.3.19 can be obtained from Ω.

Lemma 8.3.20 (Alfakih and Ye [8]). Let (G, p) be a tensegrity framework in Rr such that there exists a stress matrix Ω of (G, p) with rank Ω = n − r − 1. Then every r¯ = n − r − 1 linearly independent columns indexed by j1, . . . , jr¯ of Ω is a ˆ ˆ Gale matrix Z of p. Moreover, this Gale matrix Z has the property that zˆijk = 0 0 0 if {i, jk} ∈ E ∪ C ∪ S . Here the rows of Zˆ are indexed by {1, . . . , n} while the columns of Zˆ are indexed by j1, . . . , jr¯.

Proof. A matrix Zˆ obtained as in the lemma is obviously full column rank and 0 0 T satisﬁesz ˆijk = 0 if {i, jk} ∈ E ∪ C ∪ S . Moreover, since P Ω = 0 we have P T Zˆ = 0. Hence Zˆ is a Gale matrix of p.

We will prove the following result which is in fact stronger than Theorem 8.3.4. Theorem 8.3.21. Let (G, p) be a tensegrity framework on n vertices in Rr with r ≤ n − 2 that satisﬁes the following conditions.

1. (G, p) has a PSD proper stress matrix Ω with rank Ω = n − r − 1.

183 8.3. Universal rigidity of tensegrity frameworks

2. For each i ∈ {1, . . . , n}, the set {pj : j ∈ {i} ∪ N ∗(i)} aﬃnely spans Rr, where N ∗(i) = {j ∈ V : {i, j} ∈ B ∪ C∗ ∪ S∗}

Then (G, p) is universally rigid.

Proof. Suppose that (G, p) is a framework satisfying the conditions in the theorem. By Theorem 8.3.1 it is sufficient to show that there does not exist a tensegrity framework (G, p′) in Rr affinely-dominated by but non congruent to (G, p). By Lemma 8.3.18 it is equivalent to showing that, for a Gale matrix Z, the equation E ∗(y)Z = 1ξT has the unique solution y = 0. Let Zˆ be the Gale matrix in Lemma 8.3.20. For this matrix Zˆ, the equation E ∗(y)Zˆ = eξT is equivalent to E ∗(y)Zˆ = 0 by Lemma 8.3.19. Therefore, in the following we show that E ∗(y)Zˆ = 0 implies y = 0. ∗ j For each vertex i ∈ {1, . . . , n}, let Ji = {i} ∪ N (i). The set {p : j ∈ Ji} affinely spans Rr, for every i ∈ {1, . . . , n}, from the second assumption. Then by ˆ Corollary 8.3.12, the rows of ZJ¯i are linearly independent. Let (ˆzj)T be the jth row of Zˆ. Since E ∗(y)Zˆ = 0, for every i ∈ {1, . . . , n} n ∗ j T ∗ j T we have j=1 E (y)ij(ˆz ) = 0, which is equivalent to j∈J¯i E (y)ij(ˆz ) = 0, as ∗ E (y) = 0 for j ∈ J . The linear independence of the rows of Zˆ ¯ implies that ij P i P Ji ∗ E (y)ij = 0 for j ∈ J¯i, for every i ∈ {1, . . . , n}. Therefore y = 0. This completes the proof.

The following result indicates that, under the assumption about the stress matrix in Theorem 8.3.21, the general position condition implies the spanning condition.

Lemma 8.3.22. Let (H, p) be a tensegrity framework on n vertices in Rr which is of full dimension. Suppose that H has a stress matrix Ω of rank n−r −1. Suppose j further that, for every vertex i ∈ {1, . . . , n}, the points p for all j ∈ {i} ∪ NH (i) are in general position, where NH (i) denotes the set of neighbors of i in H. Then, j r for each i ∈ {1, . . . , n}, the set {p : j ∈ {i} ∪ NH (i)} aﬃnely spans R .

Proof. It will suﬃce to show that for each i ∈ {1, . . . , n} we have |N(i)| ≥ r + 1. Suppose by contradiction that there exists a vertex i ∈ {1, . . . , n} with at most j r r neighbors. Then since {p : j ∈ {i} ∪ NH (i)} is in general position in R , the j i vectors p − p for j ∈ NH (i) are linearly independent. On the other hand, since Ω is a stress matrix for (H, p), ω (pi − pj) = 0 holds. Therefore, ω = 0 j∈NH (i) ij ij for every j ∈ N (i). Let H′ = H − i be the graph obtained from H by deleting H P the vertex i and all its incident edges. Let p′ be obtained by restricting p on

184 Chapter 8. Universal rigidity

{1, . . . , n}\{i}. By deleting the ith row and ith column (which are all zeros) of Ω we obtain a stress matrix Ω′ for the tensegrity framework (H′, p′). This matrix Ω′ obviously has the same rank as Ω and hence rank Ω′ = n − r − 1 > n′ − r − 1. This implies that the aﬃne dimension of the tensegrity framework (H′, p′) is less than r by Lemma 8.3.9. Moreover, this aﬃne dimension can not be less than r − 1 since (G, p) is of full dimension in Rr. Therefore,

(H′, p′) is a tensegrity framework of aﬃne dimension r − 1. (8.3.20)

Since rank Ω′ = n−r−1 > 0, there must be some non-zero stress. Suppose that k is a vertex incident to an edge with a non-zero stress. Then dH′ (k) ≥ r, otherwise, the same argument as in the beginning of the proof implies that all stresses on the ′ edges incident to k in H are zero, a contradiction. Let NH′ (k) denote the set of ′ j neighbors of k in H . Then, by the assumption of the lemma,{p : j ∈ k ∪ NH′ (k)} is a set of at least r + 1 points in general position in Rr, a contradition to (8.3.20). The lemma follows.

Now we are ready to prove the main theorems.

Proof of Theorem 8.3.4

Let (G, p) be a tensegrity framework in Rr satisfying the conditions in Theorem 8.3.4. Let H = (V,B ∪C∗ ∪S∗). Then it is obvious that Ω is also a PSD stress ma- j trix for (H, p). By Lemma 8.3.22, in (H, p), for each v ∈ V , {p : j ∈ {i} ∪ NH (i)} aﬃnely spans Rr. Therefore the tensegrity framework (H, p) satisﬁes all the conditions in Theorem 8.3.21 and so (H, p) is universally rigid. Thus (G, p) is universally rigid. 2

Proof of Theorem 8.3.5

Let (G, p) be a bar framework satisfying the conditions in Theorem 8.3.5. Then (G, p), regarded as a tensegrity framework without cables and struts, obviously satiﬁes the conditions in Theorem 8.3.21. Therefore, (G, p) is universally rigid. 2

8.3.7 Counter-example and conjectures

Having Theorem 8.3.4 in hand, a natural question is whether the converse holds, that is, does every universally rigid framework in general position in Rr has a

185 8.3. Universal rigidity of tensegrity frameworks

PSD stress matrix of maximal rank. When “general” is replaced by “generic”, the answer is aﬃrmative for bar frameworks (Gortler and Thurston [45], cf. Theorem 3.2.7). It is known that the answer is also aﬃrmative for bar frameworks in general position when the underlying graphs belong to some special classes. A chordal graph is a graph without induced cycles of length greater than 3. A k-lateration graph is a graph that can be constructed from a complete graph on k vertices by recursively adding a new vertex and k edges incident to this vertex.

Theorem 8.3.23 (Alfakih [5], Alfakih, Taheri and Ye [7]). Let G be a chordal graph or an (r + 1)-lateration graph on n ≥ r + 2 vertices. If (G, p) is a framework in general position in Rr which is universally rigid then (G, p) has a PSD stress matrix of rank n − r − 1.

These results, unfortunately, can not be extended to all bar frameworks in general position. In the following we consider an example of a universally rigid bar framework in general position in R2 without any PSD stress matrices of maximal rank.

3 3 • •

6 • • 6 6 −1 −1 2 2

• 4 5 • • 4 5 • 2 •7 • 6 6 • • • 1 −1 2 1 2 (a) (b)

Figure 8.10: (a) A framework in R2 with a unique equilibrium stress. (b) A new vertex 7 is added.

Consider a framework (G, p) as in Figure 8.10(a) where the vertices 1, 2, 3 form a regular triangle, the vertices 4, 5, 6 also form a regular triangle with edges parallel to edges of 1, 2, 3 and of half edge length. Furthermore 4 (resp., 5 and 6) lies on the median from 1 (resp., 2 and 3). Note that then the distance from 4 to 1 is one-third of the median length. It is not diﬃcult to see that this framework has a unique non-zero equilibrium stress (up to a scalar multiple). Scaling this equilibrium stress so that the stress on the edge {1, 2} is −1, we can easily deduce the stress on the

186 Chapter 8. Universal rigidity other edges as indicated in the ﬁgure. The corresponding stress matrix is

4 1 1 −6 0 0  1 4 1 0 −6 0  1 1 4 0 0 −6 Ω =   . − − −   6 0 0 10 2 2    0 −6 0 −2 10 −2    0 0 −6 −2 −2 10      It is also easy to check that rank Ω = 3. The positive semideﬁniteness can also be easily checked by calculating the ﬁrst 3 leading principal minors (Lemma 2.5.2). The framework (G, p) is obviously in general position, therefore, (G, p) is a universally rigid framework in R2 by Theorem 8.3.3. Now extend (G, p) to a framework (G′, p′) in R2 by adding a new vertex 7 incident to 1, 2 and 4 such that (G′, p′) is in general position as in Figure 8.10b. Then since (G, p) is universally rigid, (G′, p′) must also be universally rigid. Sup- ′ ′ ′ pose that (ωij) is a non-zero stress matrix for (G , p ). Then one can see that ′ ′ ω1,3 must be non-zero. So we can suppose that ω1,3 = 1 = ω1,3. However, then ′ ′ ′ ′ ω3,2 = ω3,2 and ω3,6 = ω3,6 holds. Hence, ω5,6 = ω5,6 and ω4,6 = ω4,6. Thus ′ ′ again, ω5,2 = ω5,2 and ω4,5 = ω4,5. In (G, p), the equilibrium at the vertex 4 yields 4 1 4 5 4 6 ′ ′ ω4,1(p − p ) + ω4,5(p − p ) + ω4,6(p − p ) = 0. Therefore in (G , p ), the vector ′ ′4 ′5 ′ ′4 ′6 4 5 4 6 ω4,5(p − p ) + ω4,6(p − p ) = ω4,5(p − p ) + ω4,6(p − p ) lies on the same line ′4 ′1 4 1 ′ ′ ′ as p − p = p − p . Hence the equilibrium at 4 in (G , p ) implies that ω4,7 = 0 ′ as 7 does not lie on the line 1, 4. We also deduce that ω1,7 = ω2,7 = 0. So the only non-zero equilibrium stress of (G′, p′) has stress 0 on every edge incident to the vertex 7. Therefore, the corresponding stress matrix Ω′ is obtained from Ω by adding an all-zero column and an all-zero row. The rank of this matrix is obviously that of Ω which is 3 < 7 − 2 − 1.

We believe that the existence of a PSD matrix of maximal rank is necessary for bar frameworks in general position on the line, since a universally rigid framework on the line must be inﬁnitesimally rigid. Conjecture 8.3.24. Let (G, p) be a bar framework on n ≥ 3 vertices in general position in R1. If (G, p) is universally rigid then it has a PSD stress matrix of rank n − 2. We also conjecture that Theorem 3.2.7 can be extended to generic tensegrity frameworks.

187 8.3. Universal rigidity of tensegrity frameworks

Conjecture 8.3.25. Let (G, p) be a generic tensegrity framework on n vertices in Rr for some r ≤ n−2. If (G, p) is universally rigid then (G, p) has a proper stress matrix of rank n − r − 1.

188 Conclusion

Although coming into the scene with different motivations and being often dealed with different techniques, infinitesimal rigidity, global rigidity and universal rigidity are tightly related. In this thesis we have treated a wide range of problems concerning these different kinds of rigidity in various types of frameworks. Chapter 4 presented our study of local rigidity in bar-joint frameworks from a matroidal viewpoint. We characterized abstract rigidity matroids as well as intersecting submodular functions inducing these matroids. In Chapter 5, we developed combinatorial results in inductive constructions and decompositions of graphs. We provided inductive constructions of (m, d)- graded sparse graphs where different types of edges are subject to different sparsity conditions, and of (b, l)-sparse graphs where the sparsity of subgraphs depends on their vertex sets. We also obtained a decomposition of (m, d)-graded sparse graphs into graded pseudoforests. These results generalized classic results of Nash- Williams on the union of edge-disjoint spanning trees. Furthermore, we considered the directed counterpart (of the dual) of the decomposition problem and generalized a result of Edmonds to packing of matroid-based rooted arborescences. In Chapter 6, we used the results developed in Chapter 5 to obtain characterizations of the infinitesimal rigidity of several types of frameworks with mixed constraints: body-bar frameworks with bar-boundary, body-length-direction frameworks. We also investigated the effect of extension operations on direction-length graphs. Chapter 7 studied the effect of 1-extensions on the global rigidity of direction- length frameworks in general dimension, extending an earlier result of Jordán and Jackson.

189 8.3. Universal rigidity of tensegrity frameworks

Chapter 8 presented our results on universal rigidity. First we obtained a complete characterization of the universal rigidity of complete bipartite frameworks on the line and proved that every bipartite graph (except the single edge) is not generically universally rigid in any dimension. Second, we provided sufficient conditions for the universal rigidity of tensegrity frameworks (and hence bar-joint frameworks) which may be in non-general position. One of the most challenging open problem which is of great importance in rigidity theory is certainly that on the combinatorial characterization of 3-dimensional generic local and global rigidity. Though some partial results have been obtained for special classes of graphs such as molecular graphs, this problem remains difficult to tackle for the lack of good conjectures. A less studied area so far is universal rigidity. This area is particularly interesting firstly because known results are few even in dimension one. Secondly many open questions can be formed. Furthermore, to characterize universal rigidity combinatorially, even on the line, one would need a deep understanding of the motions of frameworks between different dimensions.

190 Chapter 8. Universal rigidity

Bien qu’introduites pour des motifs différents et traitées àl’aide de techniques différentes, les rigidités infinitésimale, globale et universelle sont étroitement liées. Dans cette thèse nous avons traitéune large étendue de problèmes concernant ces différentes sortes de rigidité, dans des types de charpentes variées. Le Chapitre 4 a présenténotre étude de la rigiditélocale des charpentes barres- et-joints àpartir d’un point de vue matro¨ıdal. Nous avons caractériséles matro¨ıdes de la rigiditéabstraite ainsi que les fonctions sous-modulaires intersectantes in- duisant ces matro¨ıdes. Dans le Chapitre 5, nous avons développédes résultats combinatoires sur des constructions inductives et des décompositions de graphes. Nous avons fourni des constructions inductives de graphes (m, d)-graded sparses oùdifférents types d’arêtes sont sujets àdes conditions différentes de densité, et de graphes (b, l)- sparse oùla densité des sous-graphes dépend de l’ensemble de leur sommets. Nous avons aussi obtenu une décomposition de graphes (m, d)-graded sparse en des pseudo-forêts. Ces résultats ont généralisédes résultats classiques de Nash- Williams sur les unions d’arbres couvrants àarrêtes disjointes. De plus, nous avons considéréla contrepartie orientée (du dual) du problème de décomposition et avons généraliséun résultat d’Edmonds pour le packing de matroid-based rooted arborescences. Au Chapitre 6, nous avons utiliséles résultats développés au Chapitre 5 pour obtenir la caractérisation de la rigiditéinfinitésimale de plusieurs types de charpentes avec des contraintes mixtes: charpentes body-bar avec des frontières par des barres, charpentes de body-length-direction. Nous avons aussi étudiéles effets d’opérations d’extension sur des graphes direction-length. Au Chapitre 7 nous avons étudiéles effets de 1-extensions sur la rigiditéglobale des charpentes de direction-length en dimension quelconque, étendant un résultat antérieur de Jordán et Jackson. Le Chapitre 8 a présenténos résultats sur la rigiditéuniverselle. Tout d’abord, nous avons obtenu une caractérisation complète de la rigiditéuniverselle de charpentes biparties complètes sur la ligne et avons prouvéque tout graphe biparti (exceptécelui réduit àune seule arête) n’est pas génériquement universellement rigide en toute dimension. En second lieu, nous avons fourni des conditions suff- isantes pour la rigiditéuniverselle de charpentes de tensegrité(et par conséquent de charpentes barres-et-joints) qui peuvent être en position non-générale. Un des problèmes ouverts le plus ardu qui est d’une grande importance dans la théorie de la rigiditéest certainement la caractérisation combinatoire de la rigidité

191 8.3. Universal rigidity of tensegrity frameworks générique locale et globale en dimension 3. Bien que des résultats partiels ont été obtenus pour des classes spéciales de graphes tels que les graphes moléculaires, ce problème reste difficile àrésoudre du fait du manque de bonnes conjectures. Un domaine moins étudiéjusqu’àprésent est celui de la rigidité universelle. Ce domaine est particulièrement intéressant d’une part car il y a très peu de résultats connus en dimension 1. Ensuite, de nombreuses questions ouvertes peuvent être formulées. Finalement, afin de caractériser combinatoirement la rigiditéuniverselle, même sur la ligne, la compréhension profonde des déplacements des charpentes entre des dimensions différentes est nécessaire.

192 Bibliography

[1] T. G. Abbott. Generalizations of Kempe’s universality theorem, 2008. Mas- ter’s thesis, Massachusetts Institute of Technology, Dept. of Electrical Engi- neering and Computer Science.

[2] A. Y. Alfakih. Graph rigidity via Euclidean distance matrices. Linear Algebra Appl., 310(1-3):149–165, 2000.

[3] A. Y. Alfakih. On the universal rigidity of generic bar frameworks. Contrib. Discrete Math., 5(1):7–17, 2010.

[4] A. Y. Alfakih. On bar frameworks, stress matrices and semideﬁnite programming. Math. Program., 129(1, Ser. B):113–128, 2011.

[5] A. Y. Alfakih. On stress matrices of chordal bar frameworks in general positions. 2012. arXiv/1205.3990.

[6] A. Y. Alfakih and V. H. Nguyen. On aﬃne motions and universal rigidity of tensegrity frameworks. 2013. arXiv:1305.5955.

[7] A. Y. Alfakih, N. Taheri, and Y. Ye. On stress matrices of (d + 1)-lateration frameworks in general position. Math. Program., 137(1-2, Ser. A):1–17, 2013.

[8] A. Y. Alfakih and Y. Ye. On aﬃne motions and bar frameworks in general position. Linear Algebra Appl., 438(1):31–36, 2013.

[9] L. Asimow and B. Roth. The rigidity of graphs. Trans. Amer.Math. Soc., 245:279–289, 1978.

193 Bibliography

[10] L. Asimow and B. Roth. The rigidity of graphs, II. J. Math. Anal. and Appl., 68(1):171–190, 1979.

[11] J. Bang-Jensen, A. Frank, and B. Jackson. Preserving and increasing local edge-connectivity in mixed graphs. SIAM J. Discrete Math., 8(2):155–178, 1995.

[12] K. B´erczi, S. Fujishige, and N. Kamiyama. A linear-time algorithm to ﬁnd a pair of arc-disjoint spanning in-arborescence and out-arborescence in a directed acyclic graph. Inform. Process. Lett., 109(23-24):1227–1231, 2009.

[13] A. R. Berg and T. Jord´an. A proof of Connelly’s conjecture on 3-connected circuits of the rigidity matroid. J. Combin. Theory Ser. B, 88(1):77–97, 2003.

[14] K. Bezdek and R. Connelly. Pushing disks apart—the Kneser-Poulsen conjecture in the plane. J. Reine Angew. Math., 553:221–236, 2002.

[15] P. Biswas and Y. Ye. Semideﬁnite programming for ad hoc wireless sensor network localization. In Proceedings 3rd IPSN, pages 46–54, 2004.

[16] J. Bochnak, M. Coste, and M.-F. Roy. Real algebraic geometry. Springer- Verlag, Berlin, 1998.

[17] G. Brightwell. On the complexity of diagram testing. Order, 10(4):297–303, 1993.

[18] J. Cheriyan, O. Durand de Gevigney, and Z. Szigeti. Packing of spanning rigid subgraphs and spanning trees. 2012. arXiv:1201.3727.

[19] R. Connelly. Questions, conjectures and remarks on globally rigid tensegrities. preprint, 2009, http://www.math.cornell.edu/ connelly/09- Thoughts.pdf.

[20] R. Connelly. A counterexample to the rigidity conjecture for polyhedra. Inst. Haut. Etud. Sci. Publ. Math., 47, 1977.

[21] R. Connelly. Rigidity and energy. Invent. Math., 66(1):11–33, 1982.

[22] R. Connelly. Chapter 2: Basic concepts (draft chapter for a book on rigid structures. http://www.math.cornell.edu/ connelly/rigidity.chapter.2.pdf, 1987.

194 Bibliography

[23] R. Connelly. Generic global rigidity. Discrete Comput. Geom., 33(4):549–563, 2005.

[24] R. Connelly, T. Jord´an, and W. Whiteley. Generic global rigidity of body-bar frameworks. J. Combin. Theory Ser. B, 103(6):689–705, 2013.

[25] R. Connelly and W. Whiteley. Global rigidity: the eﬀect of coning. Discrete Comput. Geom., 43(4):717–735, 2010.

[26] H. Crapo. On the generic rigidity of plane frameworks. INRIA research report, (1278), 1990.

[27] H. Crapo and W. Whiteley. Statics of frameworks and motions of panel structures, a projective geometric introduction. Structural Topology, (6):43– 82, 1982.

[28] O. Durand de Gevigney, V. H. Nguyen, and Z. Szigeti. Matroid-based packing of arborescences. SIAM J. Discrete Math., 27(1):567–574, 2013.

[29] R. Diestel. Graph Theory. Springer-Verlag, Heidelberg, 2005.

[30] R. Connelly (ed.). Conjectures and questions on global rigidity. Open problems of a mini-workshop at Cornell University, February 2011.

[31] J. Edmonds. Edge-disjoint branchings. In Combinatorial algorithms (Courant Comput. Sci. Sympos. 9, New York Univ., New York, 1972), pages 91–96. Algorithmics Press, New York, 1973.

[32] T. Eren, W. Whiteley, and P. N. Benhumeur. Further results on sensor network localization using rigidity. In Proc. of the European Conference on Wireless Sensor Networks, 2005.

[33] Z. Fekete and L. Szeg˝o. A note on [k, l]-sparse graphs. In Graph theory in Paris, Trends Math., pages 169–177. Birkh¨auser, Basel, 2007.

[34] D. C. Fisher, K. Fraughnaugh, L. Langley, and D. B. West. The number of dependent arcs in an acyclic orientation. J. Combin. Theory Ser. B, 71(1):73– 78, 1997.

[35] A. Frank. On disjoint trees and arborescences. In Algebraic Methods in Graph Theory, Colloquia Mathematica Soc. J´anos Bolyai, volume 25, pages 159–69. 1978.

195 Bibliography

[36] A. Frank. Covering branchings. Acta Sci. Math. (Szeged), 41(1-2):77–81, 1979.

[37] A. Frank. Kernel systems of directed graphs. Acta Sci. Math. (Szeged), 41(1-2):63–76, 1979.

[38] A. Frank. On the orientation of graphs. J. Combin. Theory Ser. B, 28(3):251– 261, 1980.

[39] A. Frank. Connections in combinatorial optimization. Oxford University Press, Oxford, 2011.

[40] S. Frank and J. Jiang. New classes of counterexamples to Hendrickson’s global rigidity conjecture. Discrete Comput. Geom., 45(3):574–591, April 2011.

[41] S. Fujishige. A note on disjoint arborescences. Combinatorica, 30(2):247–252, 2010.

[42] D. Gale. Neighboring vertices on a convex polyhedron. In Linear inequalities and related system, pages 255–263. Princeton University Press, 1956.

[43] H. Gluck. Almost all simply connected closed surfaces are rigid. In Geometric topology (Proc. Conf., Park City, Utah, 1974), pages 225–239. Lecture Notes in Math., Vol. 438. Springer, Berlin, 1975.

[44] S. J. Gortler, A. D. Healy, and D. P. Thurston. Characterizing generic global rigidity. Amer. J. Math., 132(4):897–939, 2010.

[45] S. J. Gortler and D. P. Thurston. Characterizing the universal rigidity of generic frameworks. 2009. arXiv:1001.0172.

[46] J. Graver. Rigidity matroids. SIAM J. Discrete Math., 4(3):355–368, 1991.

[47] J. Graver, B. Servatius, and H. Servatius. Combinatorial Rigidity. AMS, 1993.

[48] J. Graver, B. Servatius, and H. Servatius. Abstract rigidity in m-space. Contemporary Math., 178:145–151, 1994.

[49] M. Gr¨otschel, L. Lov´asz, and A. Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1(2):169–197, 1981.

196 Bibliography

[50] B. Gr¨unbaum. Convex polytopes. John Wiley & Sons, 1967.

[51] R. Haas. Characterizations of arboricity of graphs. Ars Combin., 63:129–137, 2002.

[52] R. Haas, A. Lee, I. Streinu, and L. Theran. Characterizing sparse graphs by map decompositions. J. Combin. Math. Combin. Comput., 62:3–11, 2007.

[53] B. Hendrickson. Conditions for unique graph realizations. SIAM J. Comput., 21(1):65–84, 1992.

[54] L. Henneberg. Die graphische Statik der starren Systeme. Leipzig, 1911.

[55] W. V. D. Hodge and D. Pedoe. Methods of algebraic geometry. Vol. I. Cam- bridge, 1994.

[56] D. E. Ingber. Tensegrity I. Cell structure and hierarchical systems biology. J. Cell Sci., 116:1157–1173, 2003.

[57] S. Iwata, L. Fleischer, and S. Fujishige. A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM, 48(4):761–777, 2001.

[58] B. Jackson and T. Jord´an. Connected rigidity matroids and unique realizations of graphs. J. Comb. Theory, Ser. B, 94(1):1–29, 2005.

[59] B. Jackson and T. Jord´an. On the rigidity of molecular graphs. Combina- torica, 28(6):645–658, 2008.

[60] B. Jackson and T. Jord´an. The generic rank of body-bar-and-hinge frameworks. European J. Combin., 31(2):574–588, 2010.

[61] B. Jackson and T. Jord´an. Operations preserving global rigidity of generic direction-length frameworks. Int. J. Comput. Geom. Appl., 20(6):685–706, 2010.

[62] B. Jackson, T. Jord´an, and Z. Szabadka. Globally linked pairs of vertices in equivalent realizations of graphs. Discrete Comput. Geo., 35(3):493–512, 2006.

[63] B. Jackson, T. Jord´an, and Z. Szabadka. Globally linked pairs of vertices in rigid frameworks. Technical Report TR-2012-19, Egerv´ary Research Group, Budapest, 2012. www.cs.elte.hu/egres.

197 Bibliography

[64] B. Jackson and P. Keevash. Bounded direction-length frameworks. Discrete Comput. Geom., (1):48–71, 2011.

[65] B. Jackson and P. Keevash. Necessary conditions for the global rigidity of direction-length frameworks. Discrete Comput. Geom., 46(1):72–85, 2011.

[66] B. Jackson and V. H. Nguyen. Graded sparse graphs and body-length- direction frameworks. In Proc. 8th Japanese-Hungarian Symposium on Dis- crete Mathematics and Its Applications, 2013. Veszpr´em, Hungary.

[67] D. J. Jacobs, L. A. Kuhn, and M. F. Thorpe. Flexible and rigid regions in proteins. In Rigidity Theory and Applications, pages 357–384. Kluwer, 1999.

[68] T. Jord´an. On the existence of k edge-disjoint 2-connected spanning subgraphs. J. Combin. Theory Ser. B, 95(2):257–262, 2005.

[69] T. Jord´an. Highly connected molecular graphs are rigid in three dimensions. Inform. Process. Lett., 112(8-9):356–359, 2012.

[70] T. Jord´an and V. H. Nguyen. On universally rigid frameworks on the line. Technical Report TR-2012-10, Egerv´ary Research Group, Budapest, 2012. www.cs.elte.hu/egres.

[71] N. Kamiyama, N. Katoh, and A. Takizawa. Arc-disjoint in-trees in directed graphs. Combinatorica, 29(2):197–214, 2009.

[72] N. Katoh and S. Tanigawa. A proof of the molecular conjecture. Discrete Comput. Geom., 45(4):647–700, 2011.

[73] N. Katoh and S. Tanigawa. Rooted-tree decompositions with matroid constraints and the inﬁnitesimal rigidity of frameworks with boundaries. SIAM J. Discrete Math., 27(1):155–185, 2013.

[74] Cs. Kir´aly. On maximal independent arborescence-packing. Tech- nical Report TR-2013-03, Egerv´ary Research Group, Budapest, 2013. www.cs.elte.hu/egres.

[75] G. Laman. On graphs and rigidity of plane skeletal structures. J. Eng. Math., 4:331–340, 1970.

[76] M. Laurent and A. Varvitsiotis. Positive semideﬁnite matrix completion, universal rigidity and the strong Arnold property. 2013. arXiv:1301.6616.

198 Bibliography

[77] P. D. Lax. Linear algebra. John Wiley & Sons Inc., New York, 1997.

[78] A. Lee and I. Streinu. Pebble game algorithms and sparse graphs. Discrete Math., 308(8):1425–1437, 2008.

[79] A. Lee, I. Streinu, and L. Theran. Graded sparse graphs and matroids. J.UCS, 13(11):1671–1679, 2007.

[80] L. Lov´asz and Y. Yemini. On generic ridigity in the plane. SIAM J. Algebraic Discrete Methods, 3:91–98, 1982.

[81] J. C. Maxwell. On reciprocal ﬁgures, frameworks and diagrams of forces. Transactions of the Royal Society of Edinburgh, 26:1–40, 1870.

[82] C. St. J. A. Nash-Williams. Edge-disjoint spanning trees of ﬁnite graphs. J. London Math. Soc., 36:445–450, 1961.

[83] C. St. J. A. Nash-Williams. Decomposition of ﬁnite graphs into forests. J. London Math. Soc., 39:12, 1964.

[84] J. Neˇsetˇril and V. R¨odl. More on the complexity of cover graphs. Comment. Math. Univ. Carolin., 36(2):269–278, 1995.

[85] V. H. Nguyen. On abstract rigidity matroids. SIAM J. Discrete Math., 24(2):363–369, 2010.

[86] V. H. Nguyen. 1-extensions and global rigidity of generic direction-length frameworks. Int. J. Comput. Geom. Appl., 22(6):577–591, 2012.

[87] J. Oxley. Matroid Theory. Oxford University Press, New York, 1992.

[88] K. Ratmanski. Universally rigid framework attachments. 2010. arXiv:1011.4094.

[89] A. Recski. A network theory approach to the rigidity of skeletal structures. II. Laman’s theorem and topological formulae. Discrete Appl. Math., 8(1):63–68, 1984.

[90] A. Recski. Matroid Theory and its Applications in Electric Network Theory and in Statics. Springer, Berlin, and Akad´emiai Kiad´o, Budapest, 1989.

199 Bibliography

[91] J. B. Saxe. Embeddability of weighted graphs in k-space is strongly NP- hard. Technical report, Computer Science Department, Carnegie Mellon University, 1979.

[92] A. Schrijver. A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Combin. Theory Ser. B, 80(2):346–355, 2000.

[93] A. Schrijver. Combinatorial optimization. Polyhedra and eﬃciency. Vol. B, volume 24 of Algorithms and Combinatorics. Springer-Verlag, Berlin, 2003.

[94] B. Servatius and W. Whiteley. Constraining plane conﬁgurations in CAD: combinatorics of lengths and directions. SIAM J. Discrete Math., 12:136–153, 1999.

[95] A. M.-C. So and Y. Ye. Theory of semideﬁnite programming for sensor network localization. Math. Program. Ser. B, 109:367–384, 2007.

[96] I. Streinu and L. Theran. Sparsity-certifying graph decompositions. Graphs Combin., 25(2):219–238, 2009.

[97] I. Streinu and L. Theran. Slider-pinning rigidity: a Maxwell-Laman-type theorem. Discrete Comput. Geom., 44(4):812–837, 2010.

[98] T.-S. Tay. Linking (n − 2)-dimensional panels in n-space. II. (n − 2, 2)- frameworks and body and hinge structures. Graphs Combin., 5(3):245–273, 1989.

[99] T.-S. Tay. Henneberg’s method for bar and body frameworks. Structural Topology, (17):53–58, 1991.

[100] T.-S. Tay. A new proof of Laman’s theorem. Graphs Combin., 9(4):365–370, 1993.

[101] T.-S. Tay and W. Whiteley. Recent advances in the generic rigidity of structures. Structural Topology, (9):31–38, 1984.

[102] T.-S. Tay and W. Whiteley. Generating isostatic frameworks. Structural Topology, (11):21–69, 1985.

[103] C. Thomassen. Conﬁgurations in graphs of large minimum degree, connectivity, or chromatic number. In Combinatorial Mathematics: Proceedings of the Third International Conference (New York, 1985), volume 555 of Ann. New York Acad. Sci., pages 402–412. New York Acad. Sci., New York, 1989.

200 Bibliography

[104] W. T. Tutte. On the problem of decomposing a graph into n connected factors. J. London Math. Soc., 36:221–230, 1961.

[105] S. Wells, S. Menor, B. Hespenheide, and M. F. Thorpe. Constrained geometric simulation of diﬀusive motion in proteins. Physical Biology, (2):S127– S136, 2005.

[106] W. Whiteley. Tensegrity frameworks. 1987. Unpublished manuscript.

[107] W. Whiteley. The union of matroids and the rigidity of frameworks. SIAM J. Discrete Math., 1(2):237–255, 1988.

[108] W. Whiteley. Vertex splitting in isostatic frameworks. Structural Topology, 16:23–30, 1988.

[109] W. Whiteley. Some matroids from discrete applied geometry. In Matroid theory (Seattle, WA, 1995), volume 197 of Contemp. Math., pages 171–311. Amer. Math. Soc., Providence, RI, 1996.

[110] W. Whiteley. An analogy in geometric homology: rigidity and cofac- tors on geometric graphs. In Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), volume 161 of Progr. Math., pages 413–437. Birkh¨auser Boston, Boston, MA, 1998.

[111] W. Whiteley. Counting out to the ﬂexibility of molecules. Phys. Biol., 2:116– 126, 2005.

[112] H. Whitney. On the abstract properties of linear dependence. Amer. J. Math., 57(3):509–533, 1935.

201 Abstract

The theory of rigidity studies the uniqueness of realizations of graphs, i.e., frameworks. Originally motivated by structural engineering, rigidity theory nowadays finds applications in many important problems such as predicting protein flexibility, Computer-Aided Design, sensor network localization, etc. The present thesis treats a wide range of problems concerning different kinds of rigidity, corresponding to different scopes of uniqueness (local/infinitesimal, global and universal), in various types of frameworks. First, we develop results in inductive construction and decomposition of graphs with mixed sparsity conditions as well as results on the packing of arborescences with matroidal constraints. These results are then used to obtain characterizations of infinitesimal rigidity in frameworks with mixed constraints. We also investigate the effect of extension operations on frameworks and extend a known result on the global rigidity preservation of 1-extension on direction-length frameworks in dimension two to all dimensions. For universal rigidity, where little is known, we obtain a complete characterization for the class of complete bipartite frameworks on the line. We also generalize a sufficient condition for the universal rigidity of frameworks by allowing non-general positions.

R´esum´e

La théorie de la rigiditéétudie l’unicitédes réalisations des graphes, i.e., des charpentes. Initialement motivée par l’ingénierie des structures, la théorie de la rigiditétrouve aujourd’hui des applications dans plusieurs domaines importants comme la prédiction de la flexibilitédes protéines, la conception assistée par ordinateur, la localisation dans les réseaux des capteurs, etc. Cette thèse traite une grande variétéde problèmes concernant différents types de rigidité, qui correspondent àdifférents niveaux d’unicité(locale/infinitésimale, globale et universelle) dans des modèles variés de charpentes. D’abord, nous développons des résultats sur la construction récursive et la décomposition des graphes avec des conditions mixtes de sparsitéainsi que des résultats sur le packing des arborescences avec des contraintes de matro¨ıde. Ces résultats sont alors utilisés pour obtenir des caractérisations de la rigiditéinfinitésimale des charpentes avec des contraintes mixtes. Nous étudions aussi l’effet des opérations d’extension sur des charpentes et étendons un résultat connu sur la préservation de la rigiditéglobale d’1-extension dans les charpentes àdirection et àlongueur de la dimension deux aux dimensions supérieures. Pour la rigiditéuniverselle, un sujet que l’on connait très peu, nous obtenons une caractérisation complète pour la classe des charpentes biparties complètes sur la ligne. Nous généralisons aussi une condition suffisante pour la rigidité universelle des charpentes en permettant des positions non générales.